Orientation 111

This orientation is non-standard, but related to the orientation of the Oh pointgroup, which normally would be orrientated with the C3 axes in the 111 direction. We only show one of the options of the D3d subgroups of the Oh group with orientation XYZ.

Symmetry Operations

In the D3d Point Group, with orientation 111 there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{1,1,1\}$ , $\{-1,-1,-1\}$ ,
$C_2$	$\{1,-1,0\}$ , $\{0,1,-1\}$ , $\{1,0,-1\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_6$	$\{1,1,1\}$ , $\{-1,-1,-1\}$ ,
$\sigma _d$	$\{1,-1,0\}$ , $\{0,1,-1\}$ , $\{1,0,-1\}$ ,

Different Settings

Character Table

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

Product Table

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

Sub Groups with compatible settings

Super Groups with compatible settings

Point Group Oh with orientation XYZ

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 D3d Point group with orientation 111 the form of the expansion coefficients is:

Expansion

$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -i A(2,1) & k=2\land m=-2 \\ (-1-i) A(2,1) & k=2\land m=-1 \\ (1-i) A(2,1) & k=2\land m=1 \\ i A(2,1) & k=2\land m=2 \\ \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ (-1+i) \sqrt{7} A(4,1) & k=4\land m=-3 \\ 2 i \sqrt{2} A(4,1) & k=4\land m=-2 \\ (-1-i) A(4,1) & k=4\land m=-1 \\ A(4,0) & k=4\land m=0 \\ (1-i) A(4,1) & k=4\land m=1 \\ -2 i \sqrt{2} A(4,1) & k=4\land m=2 \\ (1+i) \sqrt{7} A(4,1) & k=4\land m=3 \\ -\frac{1}{33} i \left(8 \sqrt{22} A(6,1)-\sqrt{55} B(6,2)\right) & k=6\land m=-6 \\ -\frac{(1+i) \left(A(6,1)+2 \sqrt{10} B(6,2)\right)}{\sqrt{66}} & k=6\land m=-5 \\ -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ \left(\frac{1}{6}-\frac{i}{6}\right) \left(\sqrt{10} A(6,1)-4 B(6,2)\right) & k=6\land m=-3 \\ -i B(6,2) & k=6\land m=-2 \\ (-1-i) A(6,1) & k=6\land m=-1 \\ A(6,0) & k=6\land m=0 \\ (1-i) A(6,1) & k=6\land m=1 \\ i B(6,2) & k=6\land m=2 \\ \left(-\frac{1}{6}-\frac{i}{6}\right) \left(\sqrt{10} A(6,1)-4 B(6,2)\right) & k=6\land m=3 \\ \frac{(1-i) \left(A(6,1)+2 \sqrt{10} B(6,2)\right)}{\sqrt{66}} & k=6\land m=5 \\ \frac{1}{33} i \left(8 \sqrt{22} A(6,1)-\sqrt{55} B(6,2)\right) & k=6\land m=6 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(-I)*A[2, 1], k == 2 && m == -2}, {(-1 - I)*A[2, 1], k == 2 && m == -1}, {(1 - I)*A[2, 1], k == 2 && m == 1}, {I*A[2, 1], k == 2 && m == 2}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}, {(-1 + I)*Sqrt[7]*A[4, 1], k == 4 && m == -3}, {(2*I)*Sqrt[2]*A[4, 1], k == 4 && m == -2}, {(-1 - I)*A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {(1 - I)*A[4, 1], k == 4 && m == 1}, {(-2*I)*Sqrt[2]*A[4, 1], k == 4 && m == 2}, {(1 + I)*Sqrt[7]*A[4, 1], k == 4 && m == 3}, {(-I/33)*(8*Sqrt[22]*A[6, 1] - Sqrt[55]*B[6, 2]), k == 6 && m == -6}, {((-1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6, 2]))/Sqrt[66], k == 6 && m == -5}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {(1/6 - I/6)*(Sqrt[10]*A[6, 1] - 4*B[6, 2]), k == 6 && m == -3}, {(-I)*B[6, 2], k == 6 && m == -2}, {(-1 - I)*A[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {(1 - I)*A[6, 1], k == 6 && m == 1}, {I*B[6, 2], k == 6 && m == 2}, {(-1/6 - I/6)*(Sqrt[10]*A[6, 1] - 4*B[6, 2]), k == 6 && m == 3}, {((1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6, 2]))/Sqrt[66], k == 6 && m == 5}, {(I/33)*(8*Sqrt[22]*A[6, 1] - Sqrt[55]*B[6, 2]), k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_D3d_111.Quanty

Akm = {{0, 0, A(0,0)} , 
       {2,-1, (-1+-1*I)*(A(2,1))} , 
       {2, 1, (1+-1*I)*(A(2,1))} , 
       {2,-2, (-I)*(A(2,1))} , 
       {2, 2, (I)*(A(2,1))} , 
       {4, 0, A(4,0)} , 
       {4,-1, (-1+-1*I)*(A(4,1))} , 
       {4, 1, (1+-1*I)*(A(4,1))} , 
       {4, 2, (-2*I)*((sqrt(2))*(A(4,1)))} , 
       {4,-2, (2*I)*((sqrt(2))*(A(4,1)))} , 
       {4,-3, (-1+1*I)*((sqrt(7))*(A(4,1)))} , 
       {4, 3, (1+1*I)*((sqrt(7))*(A(4,1)))} , 
       {4,-4, (sqrt(5/14))*(A(4,0))} , 
       {4, 4, (sqrt(5/14))*(A(4,0))} , 
       {6, 0, A(6,0)} , 
       {6,-1, (-1+-1*I)*(A(6,1))} , 
       {6, 1, (1+-1*I)*(A(6,1))} , 
       {6,-2, (-I)*(B(6,2))} , 
       {6, 2, (I)*(B(6,2))} , 
       {6, 3, (-1/6+-1/6*I)*((sqrt(10))*(A(6,1)) + (-4)*(B(6,2)))} , 
       {6,-3, (1/6+-1/6*I)*((sqrt(10))*(A(6,1)) + (-4)*(B(6,2)))} , 
       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6,-5, (-1+-1*I)*((1/(sqrt(66)))*(A(6,1) + (2)*((sqrt(10))*(B(6,2)))))} , 
       {6, 5, (1+-1*I)*((1/(sqrt(66)))*(A(6,1) + (2)*((sqrt(10))*(B(6,2)))))} , 
       {6,-6, (-1/33*I)*((8)*((sqrt(22))*(A(6,1))) + (-1)*((sqrt(55))*(B(6,2))))} , 
       {6, 6, (1/33*I)*((8)*((sqrt(22))*(A(6,1))) + (-1)*((sqrt(55))*(B(6,2))))} }

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) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{i \text{Asd}(2,1)}{\sqrt{5}} $	$ -\frac{(1-i) \text{Asd}(2,1)}{\sqrt{5}} $	$ 0 $	$ \frac{(1+i) \text{Asd}(2,1)}{\sqrt{5}} $	$ -\frac{i \text{Asd}(2,1)}{\sqrt{5}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ {Y_{-1}^{(1)}} $	$\color{darkred}{ 0 }$	$ \text{App}(0,0) $	$ \left(-\frac{1}{5}-\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $	$ \frac{1}{5} i \sqrt{6} \text{App}(2,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{3 i \text{Apf}(2,1)}{\sqrt{35}}+\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $	$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}}-(1-i) \sqrt{\frac{6}{35}} \text{Apf}(2,1) $	$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $	$ \frac{\left(\frac{3}{5}+\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}}-\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1) $	$ -\frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)-\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $	$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $	$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $
$ {Y_{0}^{(1)}} $	$\color{darkred}{ 0 }$	$ \left(-\frac{1}{5}+\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $	$ \text{App}(0,0) $	$ \left(\frac{1}{5}+\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $	$ i \sqrt{\frac{3}{35}} \text{Apf}(2,1)-\frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1) $	$ \left(-\frac{2}{5}+\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)-\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $	$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $	$ \left(\frac{2}{5}+\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)+\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $	$ \frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1)-i \sqrt{\frac{3}{35}} \text{Apf}(2,1) $	$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $
$ {Y_{1}^{(1)}} $	$\color{darkred}{ 0 }$	$ -\frac{1}{5} i \sqrt{6} \text{App}(2,1) $	$ \left(\frac{1}{5}-\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $	$ \text{App}(0,0) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $	$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $	$ \frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)+\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $	$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1)-\frac{\left(\frac{3}{5}-\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}} $	$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $	$ (1+i) \sqrt{\frac{6}{35}} \text{Apf}(2,1)-\frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}} $	$ -\frac{3 i \text{Apf}(2,1)}{\sqrt{35}}-\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $
$ {Y_{-2}^{(2)}} $	$ -\frac{i \text{Asd}(2,1)}{\sqrt{5}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $	$ \left(\frac{1}{21}+\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1)-\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1) $	$ \frac{2}{7} i \text{Add}(2,1)+\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $	$ \frac{5}{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_{-1}^{(2)}} $	$ -\frac{(1+i) \text{Asd}(2,1)}{\sqrt{5}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(\frac{1}{21}-\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1)-\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1) $	$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $	$ \left(-\frac{1}{7}-\frac{i}{7}\right) \text{Add}(2,1)-\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \frac{1}{7} i \sqrt{6} \text{Add}(2,1)-\frac{8}{21} i \sqrt{5} \text{Add}(4,1) $	$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ {Y_{0}^{(2)}} $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ -\frac{2}{7} i \text{Add}(2,1)-\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \left(-\frac{1}{7}+\frac{i}{7}\right) \text{Add}(2,1)-\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $	$ \left(\frac{1}{7}+\frac{i}{7}\right) \text{Add}(2,1)+\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \frac{2}{7} i \text{Add}(2,1)+\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ {Y_{1}^{(2)}} $	$ \frac{(1-i) \text{Asd}(2,1)}{\sqrt{5}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $	$ \frac{8}{21} i \sqrt{5} \text{Add}(4,1)-\frac{1}{7} i \sqrt{6} \text{Add}(2,1) $	$ \left(\frac{1}{7}-\frac{i}{7}\right) \text{Add}(2,1)+\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $	$ \left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1)-\left(\frac{1}{21}+\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ {Y_{2}^{(2)}} $	$ \frac{i \text{Asd}(2,1)}{\sqrt{5}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{5}{21} \text{Add}(4,0) $	$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $	$ -\frac{2}{7} i \text{Add}(2,1)-\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $	$ \left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1)-\left(\frac{1}{21}-\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1) $	$ \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)}} $	$\color{darkred}{ 0 }$	$ -\frac{3 i \text{Apf}(2,1)}{\sqrt{35}}-\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $	$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $	$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $	$ \left(-\frac{1}{3}-\frac{i}{3}\right) \text{Aff}(2,1)+\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)-\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $	$ \frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)+\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)+\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $	$ \left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)+\left(\frac{7}{11}-\frac{7 i}{11}\right) \text{Aff}(4,1) $	$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $	$ \left(-\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $	$ \frac{10}{429} i \sqrt{\frac{7}{33}} \left(8 \sqrt{22} \text{Aff}(6,1)-\sqrt{55} \text{Bff}(6,2)\right) $
$ {Y_{-2}^{(3)}} $	$\color{darkred}{ 0 }$	$ \frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}}-(1+i) \sqrt{\frac{6}{35}} \text{Apf}(2,1) $	$ \frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1)-i \sqrt{\frac{3}{35}} \text{Apf}(2,1) $	$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(-\frac{1}{3}+\frac{i}{3}\right) \text{Aff}(2,1)+\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)-\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $	$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $	$ -\frac{(1+i) \text{Aff}(2,1)}{\sqrt{15}}-\left(\frac{4}{33}+\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)+\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $	$ \frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}-\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)-\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $	$ \left(\frac{7}{33}-\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $	$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $	$ \left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $
$ {Y_{-1}^{(3)}} $	$\color{darkred}{ 0 }$	$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $	$ \left(-\frac{2}{5}-\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)-\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $	$ -\frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)-\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ -\frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)-\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $	$ -\frac{(1-i) \text{Aff}(2,1)}{\sqrt{15}}-\left(\frac{4}{33}-\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)+\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $	$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $	$ \left(-\frac{1}{15}-\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)-\left(\frac{25}{429}+\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $	$ \frac{2}{5} i \sqrt{\frac{2}{3}} \text{Aff}(2,1)-\frac{8}{33} i \sqrt{5} \text{Aff}(4,1)+\frac{10}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ \left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)-\left(\frac{7}{33}-\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1) $	$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $
$ {Y_{0}^{(3)}} $	$\color{darkred}{ 0 }$	$ \frac{\left(\frac{3}{5}-\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}}-\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1) $	$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $	$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1)-\frac{\left(\frac{3}{5}+\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)+\left(\frac{7}{11}+\frac{7 i}{11}\right) \text{Aff}(4,1) $	$ -\frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}+\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)+\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $	$ \left(-\frac{1}{15}+\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)-\left(\frac{25}{429}-\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $	$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $	$ \left(\frac{1}{15}+\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)+\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)+\left(\frac{25}{429}+\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $	$ \frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}-\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)-\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $	$ \left(-\frac{7}{11}+\frac{7 i}{11}\right) \text{Aff}(4,1)-\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $
$ {Y_{1}^{(3)}} $	$\color{darkred}{ 0 }$	$ \frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)+\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $	$ \left(\frac{2}{5}-\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)+\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $	$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $	$ \left(\frac{7}{33}+\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $	$ -\frac{2}{5} i \sqrt{\frac{2}{3}} \text{Aff}(2,1)+\frac{8}{33} i \sqrt{5} \text{Aff}(4,1)-\frac{10}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ \left(\frac{1}{15}-\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)+\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)+\left(\frac{25}{429}-\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $	$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $	$ \frac{(1+i) \text{Aff}(2,1)}{\sqrt{15}}+\left(\frac{4}{33}+\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $	$ \frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)+\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)+\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $
$ {Y_{2}^{(3)}} $	$\color{darkred}{ 0 }$	$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $	$ i \sqrt{\frac{3}{35}} \text{Apf}(2,1)-\frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1) $	$ (1-i) \sqrt{\frac{6}{35}} \text{Apf}(2,1)-\frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(-\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $	$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $	$ \left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)-\left(\frac{7}{33}+\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1) $	$ -\frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}+\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)+\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $	$ \frac{(1-i) \text{Aff}(2,1)}{\sqrt{15}}+\left(\frac{4}{33}-\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $	$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $	$ \left(\frac{1}{3}+\frac{i}{3}\right) \text{Aff}(2,1)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)+\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $
$ {Y_{3}^{(3)}} $	$\color{darkred}{ 0 }$	$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $	$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $	$ \frac{3 i \text{Apf}(2,1)}{\sqrt{35}}+\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ -\frac{10}{429} i \sqrt{\frac{7}{33}} \left(8 \sqrt{22} \text{Aff}(6,1)-\sqrt{55} \text{Bff}(6,2)\right) $	$ \left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $	$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $	$ \left(-\frac{7}{11}-\frac{7 i}{11}\right) \text{Aff}(4,1)-\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $	$ -\frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)-\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $	$ \left(\frac{1}{3}-\frac{i}{3}\right) \text{Aff}(2,1)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)+\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $	$ \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+y+z} $	$\color{darkred}{ 0 }$	$ \frac{1+i}{\sqrt{6}} $	$ \frac{1}{\sqrt{3}} $	$ -\frac{1-i}{\sqrt{6}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ p_{x-y} $	$\color{darkred}{ 0 }$	$ \frac{1}{2}-\frac{i}{2} $	$ 0 $	$ -\frac{1}{2}-\frac{i}{2} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ p_{3z-r} $	$\color{darkred}{ 0 }$	$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $	$ \sqrt{\frac{2}{3}} $	$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ d_{\text{yz}+\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_{\text{yz}-\text{xz}} $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ -\frac{1}{2}+\frac{i}{2} $	$ 0 $	$ \frac{1}{2}+\frac{i}{2} $	$ 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 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \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_{x^2-y^2} $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{1}{\sqrt{2}} $	$ 0 $	$ 0 $	$ 0 $	$ \frac{1}{\sqrt{2}} $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ d_{3z^2-r^2} $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 1 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ f_{\text{xyz}} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ \frac{i}{\sqrt{2}} $	$ 0 $	$ 0 $	$ 0 $	$ -\frac{i}{\sqrt{2}} $	$ 0 $
$ f_{x^3+y^3+z^3} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $	$ 0 $	$ -\frac{1}{4}-\frac{i}{4} $	$ \frac{1}{\sqrt{3}} $	$ \frac{1}{4}-\frac{i}{4} $	$ 0 $	$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $
$ f_{x^3-y^3} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $	$ 0 $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $	$ 0 $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $
$ f_{2z^3-x^3-y^3} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $	$ 0 $	$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $	$ \sqrt{\frac{2}{3}} $	$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $
$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ -\frac{1}{4}-\frac{i}{4} $	$ \frac{1}{\sqrt{6}} $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $	$ \frac{1}{\sqrt{6}} $	$ \frac{1}{4}-\frac{i}{4} $
$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $	$ \frac{1}{\sqrt{3}} $	$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $	$ 0 $	$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $	$ \frac{1}{\sqrt{3}} $	$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $
$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $	$ 0 $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $	$ 0 $	$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $

One particle coupling on a basis of symmetry adapted functions

After rotation we find

$ $	$ \text{s} $	$ p_{x+y+z} $	$ p_{x-y} $	$ p_{3z-r} $	$ d_{\text{yz}+\text{xz}+\text{xy}} $	$ d_{\text{yz}-\text{xz}} $	$ d_{2\text{xy}-\text{xz}-\text{yz}} $	$ d_{x^2-y^2} $	$ d_{3z^2-r^2} $	$ f_{\text{xyz}} $	$ f_{x^3+y^3+z^3} $	$ f_{x^3-y^3} $	$ f_{2z^3-x^3-y^3} $	$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $
$ \text{s} $	$ \text{Ass}(0,0) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ -\sqrt{\frac{6}{5}} \text{Asd}(2,1) $	$ 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+y+z} $	$\color{darkred}{ 0 }$	$ \text{App}(0,0)-\frac{2}{5} \sqrt{6} \text{App}(2,1) $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \frac{8 \text{Apf}(4,1)}{\sqrt{21}}-3 \sqrt{\frac{2}{35}} \text{Apf}(2,1) $	$ \frac{6}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}-\frac{4}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ p_{x-y} $	$\color{darkred}{ 0 }$	$ 0 $	$ \text{App}(0,0)+\frac{1}{5} \sqrt{6} \text{App}(2,1) $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $	$ 0 $	$ 0 $	$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $	$ 0 $
$ p_{3z-r} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ \text{App}(0,0)+\frac{1}{5} \sqrt{6} \text{App}(2,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $	$ 0 $	$ 0 $	$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $
$ d_{\text{yz}+\text{xz}+\text{xy}} $	$ -\sqrt{\frac{6}{5}} \text{Asd}(2,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \text{Add}(0,0)-\frac{2}{7} \sqrt{6} \text{Add}(2,1)-\frac{4}{21} \text{Add}(4,0)+\frac{16}{21} \sqrt{5} \text{Add}(4,1) $	$ 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}} $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ \text{Add}(0,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,1)-\frac{4}{21} \text{Add}(4,0)-\frac{8}{21} \sqrt{5} \text{Add}(4,1) $	$ 0 $	$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $	$ 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 $	$ \text{Add}(0,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,1)-\frac{4}{21} \text{Add}(4,0)-\frac{8}{21} \sqrt{5} \text{Add}(4,1) $	$ 0 $	$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$ d_{x^2-y^2} $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $	$ 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 $	$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $	$ 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_{\text{xyz}} $	$\color{darkred}{ 0 }$	$ \frac{8 \text{Apf}(4,1)}{\sqrt{21}}-3 \sqrt{\frac{2}{35}} \text{Apf}(2,1) $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ \text{Aff}(0,0)-\frac{4}{11} \text{Aff}(4,0)+\frac{80}{143} \text{Aff}(6,0) $	$ 2 \sqrt{\frac{2}{15}} \text{Aff}(2,1)-\frac{4}{11} \text{Aff}(4,1)-\frac{40}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ f_{x^3+y^3+z^3} $	$\color{darkred}{ 0 }$	$ \frac{6}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}-\frac{4}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $	$ 0 $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 2 \sqrt{\frac{2}{15}} \text{Aff}(2,1)-\frac{4}{11} \text{Aff}(4,1)-\frac{40}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $	$ \text{Aff}(0,0)+\frac{1}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,1)+\frac{2}{11} \text{Aff}(4,0)+\frac{8}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)+\frac{100}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{20}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ f_{x^3-y^3} $	$\color{darkred}{ 0 }$	$ 0 $	$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $	$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $	$ 0 $
$ f_{2z^3-x^3-y^3} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $	$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $
$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$\color{darkred}{ 0 }$	$ 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)+\sqrt{\frac{2}{3}} \text{Aff}(2,1)-\frac{2}{33} \text{Aff}(4,0)+\frac{8}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{20}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $
$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$\color{darkred}{ 0 }$	$ 0 $	$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $	$ 0 $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $	$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $	$ 0 $
$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$ 0 $	$ 0 $	$ 0 $	$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $	$ 0 $	$ 0 $	$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $

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_D3d_111.Quanty.nb

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

Input format suitable for Quanty

Akm_D3d_111.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

$$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\ \frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=-2 \\ \frac{\left(\frac{5}{3}+\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=-1 \\ -\frac{\left(\frac{5}{3}-\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=1 \\ -\frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=2 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(((5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == -2}, {((5/3 + (5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == -1}, {((-5/3 + (5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == 1}, {(((-5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == 2}}, 0]

Input format suitable for Quanty

Akm_D3d_111.Quanty

Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , 
       {2, 1, (-5/3+5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
       {2,-1, (5/3+5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
       {2, 2, (-5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , 
       {2,-2, (5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} }

The Hamiltonian on a basis of spherical Harmonics

$ $	$ {Y_{-1}^{(1)}} $	$ {Y_{0}^{(1)}} $	$ {Y_{1}^{(1)}} $
$ {Y_{-1}^{(1)}} $	$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $	$ \frac{1}{3} \sqrt[4]{-1} (\text{Ea2u}-\text{Eeu}) $	$ -\frac{1}{3} i (\text{Ea2u}-\text{Eeu}) $
$ {Y_{0}^{(1)}} $	$ \frac{1}{3} (-1)^{3/4} (\text{Eeu}-\text{Ea2u}) $	$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $	$ \frac{1}{3} \sqrt[4]{-1} (\text{Eeu}-\text{Ea2u}) $
$ {Y_{1}^{(1)}} $	$ \frac{1}{3} i (\text{Ea2u}-\text{Eeu}) $	$ \frac{1}{3} (-1)^{3/4} (\text{Ea2u}-\text{Eeu}) $	$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $

The Hamiltonian on a basis of symmetric functions

$ $	$ p_{x+y+z} $	$ p_{x-y} $	$ p_{3z-r} $
$ p_{x+y+z} $	$ \text{Ea2u} $	$ 0 $	$ 0 $
$ p_{x-y} $	$ 0 $	$ \text{Eeu} $	$ 0 $
$ p_{3z-r} $	$ 0 $	$ 0 $	$ \text{Eeu} $

Rotation matrix used

$ $	$ {Y_{-1}^{(1)}} $	$ {Y_{0}^{(1)}} $	$ {Y_{1}^{(1)}} $
$ p_{x+y+z} $	$ \frac{1+i}{\sqrt{6}} $	$ \frac{1}{\sqrt{3}} $	$ -\frac{1-i}{\sqrt{6}} $
$ p_{x-y} $	$ \frac{1}{2}-\frac{i}{2} $	$ 0 $	$ -\frac{1}{2}-\frac{i}{2} $
$ p_{3z-r} $	$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $	$ \sqrt{\frac{2}{3}} $	$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $

Irriducible representations and their onsite energy

	$$\text{Ea2u}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta )}{2 \sqrt{\pi }}$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{x+y+z}{2 \sqrt{\pi }}$$
	$$\text{Eeu}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{3}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi ))$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{3}{2 \pi }} (x-y)$$
	$$\text{Eeu}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$-\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta )}{2 \sqrt{2 \pi }}$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$-\frac{x+y-2 z}{2 \sqrt{2 \pi }}$$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\ \frac{1}{6} i \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg}\pi -4 \sqrt{3} \text{Meg}\right) & k=2\land m=-2 \\ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg}\pi -4 \sqrt{3} \text{Meg}\right) & k=2\land m=-1 \\ \left(\frac{1}{6}-\frac{i}{6}\right) \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg}\pi +4 \sqrt{3} \text{Meg}\right) & k=2\land m=1 \\ \frac{1}{6} i \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg}\pi +4 \sqrt{3} \text{Meg}\right) & k=2\land m=2 \\ -\frac{1}{2} \sqrt{\frac{7}{10}} (\text{Ea1g}+2 \text{Eeg}\pi -3 \text{Eeg}\sigma ) & k=4\land (m=-4\lor m=4) \\ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right) & k=4\land m=-3 \\ \frac{i \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=-2 \\ -\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=-1 \\ -\frac{7}{10} (\text{Ea1g}+2 \text{Eeg}\pi -3 \text{Eeg}\sigma ) & k=4\land m=0 \\ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=1 \\ -\frac{i \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=2 \\ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg}\pi +3 \sqrt{2} \text{Meg}\right) & k=4\land m=3 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1g + 2*(Eeg\[Pi] + Eeg\[Sigma]))/5, k == 0 && m == 0}, {(I/6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg), k == 2 && m == -2}, {(1/6 + I/6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg), k == 2 && m == -1}, {(1/6 - I/6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg), k == 2 && m == 1}, {(I/6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg), k == 2 && m == 2}, {-(Sqrt[7/10]*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/2, k == 4 && (m == -4 || m == 4)}, {(-1/4 + I/4)*Sqrt[7/5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg), k == 4 && m == -3}, {(I*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[10], k == 4 && m == -2}, {((-1/4 - I/4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[5], k == 4 && m == -1}, {(-7*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/10, k == 4 && m == 0}, {((1/4 - I/4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[5], k == 4 && m == 1}, {((-I)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[10], k == 4 && m == 2}, {(1/4 + I/4)*Sqrt[7/5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg), k == 4 && m == 3}}, 0]

Input format suitable for Quanty

Akm_D3d_111.Quanty

Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg\[Pi] + Eeg\[Sigma]))} , 
       {2, 1, (1/6+-1/6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , 
       {2,-1, (1/6+1/6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , 
       {2, 2, (1/6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , 
       {2,-2, (1/6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , 
       {4, 0, (-7/10)*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma]))} , 
       {4,-1, (-1/4+-1/4*I)*((1/(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4, 1, (1/4+-1/4*I)*((1/(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4, 2, (-I)*((1/(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4,-2, (I)*((1/(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4,-3, (-1/4+1/4*I)*((sqrt(7/5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4, 3, (1/4+1/4*I)*((sqrt(7/5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , 
       {4,-4, (-1/2)*((sqrt(7/10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} , 
       {4, 4, (-1/2)*((sqrt(7/10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} }

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}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $	$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) $	$ \frac{i \text{Meg}}{\sqrt{3}} $	$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $	$ \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $
$ {Y_{-1}^{(2)}} $	$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $	$ \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) $	$ \text{Meg} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right] $	$ -\frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) $	$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $
$ {Y_{0}^{(2)}} $	$ -\frac{i \text{Meg}}{\sqrt{3}} $	$ \frac{(-1)^{3/4} \text{Meg}}{\sqrt{6}} $	$ \text{Eeg$\sigma $} $	$ \frac{\sqrt[4]{-1} \text{Meg}}{\sqrt{6}} $	$ \frac{i \text{Meg}}{\sqrt{3}} $
$ {Y_{1}^{(2)}} $	$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $	$ \frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) $	$ \text{Meg} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right] $	$ \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) $	$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $
$ {Y_{2}^{(2)}} $	$ \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $	$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) $	$ -\frac{i \text{Meg}}{\sqrt{3}} $	$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $	$ \frac{1}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $

The Hamiltonian on a basis of symmetric functions

$ $	$ d_{\text{yz}+\text{xz}+\text{xy}} $	$ d_{\text{yz}-\text{xz}} $	$ d_{2\text{xy}-\text{xz}-\text{yz}} $	$ d_{x^2-y^2} $	$ d_{3z^2-r^2} $
$ d_{\text{yz}+\text{xz}+\text{xy}} $	$ \text{Ea1g} $	$ 0 $	$ 0 $	$ 0 $	$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Meg} \left(\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(1+i)\right)}{\sqrt{2}} $
$ d_{\text{yz}-\text{xz}} $	$ 0 $	$ \text{Eeg$\pi $} $	$ 0 $	$ \text{Meg} $	$ \left(-\frac{1}{2}-\frac{i}{2}\right) \text{Meg} \left(\text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]\right) $
$ d_{2\text{xy}-\text{xz}-\text{yz}} $	$ 0 $	$ 0 $	$ \text{Eeg$\pi $} $	$ 0 $	$ \left(\frac{1}{6}+\frac{i}{6}\right) \text{Meg} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(2-2 i)\right) $
$ d_{x^2-y^2} $	$ 0 $	$ \text{Meg} $	$ 0 $	$ \text{Eeg$\sigma $} $	$ 0 $
$ d_{3z^2-r^2} $	$ 0 $	$ 0 $	$ \text{Meg} $	$ 0 $	$ \text{Eeg$\sigma $} $

Rotation matrix used

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

Irriducible representations and their onsite energy

	$$\text{Ea1g}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$$
	$$\text{Eeg$\pi $}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{4} \sqrt{\frac{15}{2 \pi }} \sin (2 \theta ) (\sin (\phi )-\cos (\phi ))$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{15}{2 \pi }} z (y-x)$$
	$$\text{Eeg$\pi $}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$$
	$$\text{Eeg$\sigma $}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$
	$$\text{Eeg$\sigma $}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 (\text{Eeu1}+\text{Eeu2})) & k=0\land m=0 \\ -\frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=-2 \\ -\frac{\left(\frac{5}{28}+\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=-1 \\ \frac{\left(\frac{5}{28}-\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=1 \\ \frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=2 \\ -\frac{1}{4} \sqrt{\frac{5}{14}} (\text{Ea1u}+6 \text{Ea2u1}-3 \text{Ea2u2}-6 \text{Eeu1}+2 \text{Eeu2}) & k=4\land (m=-4\lor m=4) \\ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=-3 \\ \frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=-2 \\ \left(-\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=-1 \\ \frac{1}{4} (-\text{Ea1u}-6 \text{Ea2u1}+3 \text{Ea2u2}+6 \text{Eeu1}-2 \text{Eeu2}) & k=4\land m=0 \\ \left(\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=1 \\ -\frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=2 \\ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=3 \\ \frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=-6 \\ \left(-\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=-5 \\ \frac{13 (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ -\frac{\left(\frac{13}{40}-\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=-3 \\ -\frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=-2 \\ \frac{\left(\frac{13}{20}+\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=-1 \\ -\frac{13}{280} (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2}) & k=6\land m=0 \\ -\frac{\left(\frac{13}{20}-\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=1 \\ \frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=2 \\ \frac{\left(\frac{13}{40}+\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=3 \\ \left(\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=5 \\ -\frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=6 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*(Eeu1 + Eeu2))/7, k == 0 && m == 0}, {(((-5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == -2}, {((-5/28 - (5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == -1}, {((5/28 - (5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == 1}, {(((5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == 2}, {-(Sqrt[5/14]*(Ea1u + 6*Ea2u1 - 3*Ea2u2 - 6*Eeu1 + 2*Eeu2))/4, k == 4 && (m == -4 || m == 4)}, {((-1/4 + I/4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[7], k == 4 && m == -3}, {((I/7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[2], k == 4 && m == -2}, {(-1/28 - I/28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == -1}, {(-Ea1u - 6*Ea2u1 + 3*Ea2u2 + 6*Eeu1 - 2*Eeu2)/4, k == 4 && m == 0}, {(1/28 - I/28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == 1}, {((-I/7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[2], k == 4 && m == 2}, {((1/4 + I/4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[7], k == 4 && m == 3}, {((13*I)/80)*Sqrt[11/21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu), k == 6 && m == -6}, {(-13/40 - (13*I)/40)*Sqrt[11/7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == -5}, {(13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {((-13/40 + (13*I)/40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/Sqrt[21], k == 6 && m == -3}, {(((-13*I)/80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/Sqrt[21], k == 6 && m == -2}, {((13/20 + (13*I)/20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/Sqrt[42], k == 6 && m == -1}, {(-13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/280, k == 6 && m == 0}, {((-13/20 + (13*I)/20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/Sqrt[42], k == 6 && m == 1}, {(((13*I)/80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/Sqrt[21], k == 6 && m == 2}, {((13/40 + (13*I)/40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/Sqrt[21], k == 6 && m == 3}, {(13/40 - (13*I)/40)*Sqrt[11/7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == 5}, {((-13*I)/80)*Sqrt[11/21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu), k == 6 && m == 6}}, 0]

Input format suitable for Quanty

Akm_D3d_111.Quanty

Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1 + Eeu2))} , 
       {2,-1, (-5/28+-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {2, 1, (5/28+-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {2,-2, (-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {2, 2, (5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , 
       {4, 0, (1/4)*((-1)*(Ea1u) + (-6)*(Ea2u1) + (3)*(Ea2u2) + (6)*(Eeu1) + (-2)*(Eeu2))} , 
       {4,-1, (-1/28+-1/28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , 
       {4, 1, (1/28+-1/28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , 
       {4, 2, (-1/7*I)*((1/(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4,-2, (1/7*I)*((1/(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4,-3, (-1/4+1/4*I)*((1/(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4, 3, (1/4+1/4*I)*((1/(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , 
       {4,-4, (-1/4)*((sqrt(5/14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , 
       {4, 4, (-1/4)*((sqrt(5/14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , 
       {6, 0, (-13/280)*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2))} , 
       {6, 1, (-13/20+13/20*I)*((1/(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , 
       {6,-1, (13/20+13/20*I)*((1/(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , 
       {6,-2, (-13/80*I)*((1/(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , 
       {6, 2, (13/80*I)*((1/(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , 
       {6,-3, (-13/40+13/40*I)*((1/(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , 
       {6, 3, (13/40+13/40*I)*((1/(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , 
       {6,-4, (13/40)*((1/(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , 
       {6, 4, (13/40)*((1/(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , 
       {6,-5, (-13/40+-13/40*I)*((sqrt(11/7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , 
       {6, 5, (13/40+-13/40*I)*((sqrt(11/7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , 
       {6, 6, (-13/80*I)*((sqrt(11/21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} , 
       {6,-6, (13/80*I)*((sqrt(11/21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} }

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}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Meu}-\text{Ma2u})\right)}{\sqrt{6}} $	$ \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $	$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $	$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $	$ -\frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $
$ {Y_{-2}^{(3)}} $	$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $	$ \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) $	$ -\frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $	$ \frac{i \text{Ma2u}}{\sqrt{6}} $	$ \frac{1}{12} (-1)^{3/4} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) $	$ \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $
$ {Y_{-1}^{(3)}} $	$ -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $	$ \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $	$ \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $	$ \frac{1}{24} i \left(5 \text{Ea1u}-3 \text{Ea2u2}+3 \text{Eeu1}-5 \text{Eeu2}-6 \sqrt{5} \text{Meu}\right) $	$ \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) $	$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $
$ {Y_{0}^{(3)}} $	$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $	$ -\frac{i \text{Ma2u}}{\sqrt{6}} $	$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $	$ \frac{1}{3} (\text{Ea2u2}+2 \text{Eeu1}) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea2u2}-\text{Eeu1}-\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $	$ \frac{i \text{Ma2u}}{\sqrt{6}} $	$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $
$ {Y_{1}^{(3)}} $	$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $	$ \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) $	$ \frac{1}{24} i \left(-5 \text{Ea1u}+3 \text{Ea2u2}-3 \text{Eeu1}+5 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $	$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $	$ \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) $	$ \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $	$ \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $
$ {Y_{2}^{(3)}} $	$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $	$ \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) $	$ \frac{1}{12} \sqrt[4]{-1} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) $	$ -\frac{i \text{Ma2u}}{\sqrt{6}} $	$ -\frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $	$ \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $
$ {Y_{3}^{(3)}} $	$ \frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $	$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $	$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $	$ \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $	$ -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $	$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $	$ \frac{1}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) $

The Hamiltonian on a basis of symmetric functions

$ $	$ f_{\text{xyz}} $	$ f_{x^3+y^3+z^3} $	$ f_{x^3-y^3} $	$ f_{2z^3-x^3-y^3} $	$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $
$ f_{\text{xyz}} $	$ \text{Ea2u1} $	$ \text{Ma2u} $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ f_{x^3+y^3+z^3} $	$ \text{Ma2u} $	$ \text{Ea2u2} $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ f_{x^3-y^3} $	$ 0 $	$ 0 $	$ \text{Eeu1} $	$ 0 $	$ 0 $	$ \text{Meu} $	$ 0 $
$ f_{2z^3-x^3-y^3} $	$ 0 $	$ 0 $	$ 0 $	$ \text{Eeu1} $	$ 0 $	$ 0 $	$ \text{Meu} $
$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ \text{Ea1u} $	$ 0 $	$ 0 $
$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$ 0 $	$ 0 $	$ \text{Meu} $	$ 0 $	$ 0 $	$ \text{Eeu2} $	$ 0 $
$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $	$ 0 $	$ 0 $	$ 0 $	$ \text{Meu} $	$ 0 $	$ 0 $	$ \text{Eeu2} $

Rotation matrix used

$ $	$ {Y_{-3}^{(3)}} $	$ {Y_{-2}^{(3)}} $	$ {Y_{-1}^{(3)}} $	$ {Y_{0}^{(3)}} $	$ {Y_{1}^{(3)}} $	$ {Y_{2}^{(3)}} $	$ {Y_{3}^{(3)}} $
$ f_{\text{xyz}} $	$ 0 $	$ \frac{i}{\sqrt{2}} $	$ 0 $	$ 0 $	$ 0 $	$ -\frac{i}{\sqrt{2}} $	$ 0 $
$ f_{x^3+y^3+z^3} $	$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $	$ 0 $	$ -\frac{1}{4}-\frac{i}{4} $	$ \frac{1}{\sqrt{3}} $	$ \frac{1}{4}-\frac{i}{4} $	$ 0 $	$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $
$ f_{x^3-y^3} $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $	$ 0 $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $	$ 0 $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $
$ f_{2z^3-x^3-y^3} $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $	$ 0 $	$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $	$ \sqrt{\frac{2}{3}} $	$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $
$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$ -\frac{1}{4}-\frac{i}{4} $	$ \frac{1}{\sqrt{6}} $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $	$ \frac{1}{\sqrt{6}} $	$ \frac{1}{4}-\frac{i}{4} $
$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $	$ \frac{1}{\sqrt{3}} $	$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $	$ 0 $	$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $	$ \frac{1}{\sqrt{3}} $	$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $
$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $	$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $	$ 0 $	$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $	$ 0 $	$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $	$ 0 $	$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $

Irriducible representations and their onsite energy

	$$\text{Ea2u1}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$
	$$\text{Ea2u2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\left(\frac{1}{32}+\frac{i}{32}\right) \sqrt{\frac{7}{3 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{16} \sqrt{\frac{7}{3 \pi }} \left(5 x^3-15 x^2 y-3 x \left(5 y^2+5 z^2-1\right)+5 y^3+y \left(3-15 z^2\right)+4 z \left(5 z^2-3\right)\right)$$
	$$\text{Eeu1}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{8} \sqrt{\frac{7}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(5 \sin ^2(\theta ) \sin (2 \phi )-5 \cos (2 \theta )-1\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{16} \sqrt{\frac{7}{2 \pi }} (x-y) \left(5 x^2+20 x y+5 y^2-15 z^2+3\right)$$
	$$\text{Eeu1}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\left(-\frac{1}{32}-\frac{i}{32}\right) \sqrt{\frac{7}{6 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )-(4-4 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{16} \sqrt{\frac{7}{6 \pi }} \left(-5 x^3+15 x^2 y+3 x \left(5 y^2+5 z^2-1\right)-5 y^3+3 y \left(5 z^2-1\right)+8 z \left(5 z^2-3\right)\right)$$
	$$\text{Ea1u}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(-\sin ^2(\theta ) \sin (2 \phi )+\sin (2 \theta ) (\sin (\phi )+\cos (\phi ))-2 \cos ^2(\theta )\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} (x-y) \left(x^2+4 x (y-z)+y^2-4 y z+5 z^2-1\right)$$
	$$\text{Eeu2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{8} \sqrt{\frac{35}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )+2 \sin (2 \theta ) (\sin (\phi )+\cos (\phi ))+2 \cos ^2(\theta )\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{16} \sqrt{\frac{35}{2 \pi }} (x-y) \left(x^2+4 x (y+2 z)+y^2+8 y z+5 z^2-1\right)$$
	$$\text{Eeu2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$	$$-\frac{1}{8} \sqrt{\frac{105}{2 \pi }} \sin (\theta ) (\sin (\phi )+\cos (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )-2 \cos ^2(\theta )\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$	$$\frac{1}{16} \sqrt{\frac{105}{2 \pi }} (x+y) \left(x^2-4 x y+y^2+5 z^2-1\right)$$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} 0 & k=0\land m=0 \\ i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-2 \\ (1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-1 \\ (-1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=1 \\ -i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=2 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {I*Sqrt[5/6]*Ma1g, k == 2 && m == -2}, {(1 + I)*Sqrt[5/6]*Ma1g, k == 2 && m == -1}, {(-1 + I)*Sqrt[5/6]*Ma1g, k == 2 && m == 1}, {(-I)*Sqrt[5/6]*Ma1g, k == 2 && m == 2}}, 0]

Input format suitable for Quanty

Akm_D3d_111.Quanty

Akm = {{2, 1, (-1+1*I)*((sqrt(5/6))*(Ma1g))} , 
       {2,-1, (1+1*I)*((sqrt(5/6))*(Ma1g))} , 
       {2, 2, (-I)*((sqrt(5/6))*(Ma1g))} , 
       {2,-2, (I)*((sqrt(5/6))*(Ma1g))} }

The Hamiltonian on a basis of spherical Harmonics

$ $	$ {Y_{-2}^{(2)}} $	$ {Y_{-1}^{(2)}} $	$ {Y_{0}^{(2)}} $	$ {Y_{1}^{(2)}} $	$ {Y_{2}^{(2)}} $
$ {Y_{0}^{(0)}} $	$ -\frac{i \text{Ma1g}}{\sqrt{6}} $	$ \frac{(1-i) \text{Ma1g}}{\sqrt{6}} $	$ 0 $	$ -\frac{(1+i) \text{Ma1g}}{\sqrt{6}} $	$ \frac{i \text{Ma1g}}{\sqrt{6}} $

The Hamiltonian on a basis of symmetric functions

$ $	$ d_{\text{yz}+\text{xz}+\text{xy}} $	$ d_{\text{yz}-\text{xz}} $	$ d_{2\text{xy}-\text{xz}-\text{yz}} $	$ d_{x^2-y^2} $	$ d_{3z^2-r^2} $
$ \text{s} $	$ \text{Ma1g} $	$ 0 $	$ 0 $	$ 0 $	$ 0 $

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ \frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-2 \\ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-1 \\ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=1 \\ -\frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=2 \\ -\frac{1}{4} \sqrt{\frac{3}{10}} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land (m=-4\lor m=4) \\ \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-3 \\ i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-2 \\ \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-1 \\ -\frac{1}{20} \sqrt{21} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land m=0 \\ \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=1 \\ -i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=2 \\ \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & \text{True} \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_111.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -2}, {(1/3 + I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -1}, {(-1/3 + I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 1}, {(-I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 2}, {-(Sqrt[3/10]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/4, k == 4 && (m == -4 || m == 4)}, {(-1/2 + I/2)*Sqrt[3]*(Ma2u1 - Meu2), k == 4 && m == -3}, {I*Sqrt[6/7]*(Ma2u1 - Meu2), k == 4 && m == -2}, {(-1/2 - I/2)*Sqrt[3/7]*(Ma2u1 - Meu2), k == 4 && m == -1}, {-(Sqrt[21]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/20, k == 4 && m == 0}, {(1/2 - I/2)*Sqrt[3/7]*(Ma2u1 - Meu2), k == 4 && m == 1}, {(-I)*Sqrt[6/7]*(Ma2u1 - Meu2), k == 4 && m == 2}}, (1/2 + I/2)*Sqrt[3]*(Ma2u1 - Meu2)]

Input format suitable for Quanty

Akm_D3d_111.Quanty

Akm = {{2, 1, (-1/3+1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {2,-1, (1/3+1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {2, 2, (-1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {2,-2, (1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , 
       {4, 0, (-1/20)*((sqrt(21))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , 
       {4,-1, (-1/2+-1/2*I)*((sqrt(3/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4, 1, (1/2+-1/2*I)*((sqrt(3/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4, 2, (-I)*((sqrt(6/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4,-2, (I)*((sqrt(6/7))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4,-3, (-1/2+1/2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4, 3, (1/2+1/2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , 
       {4,-4, (-1/4)*((sqrt(3/10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , 
       {4, 4, (-1/4)*((sqrt(3/10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} }

The Hamiltonian on a basis of spherical Harmonics

$ $	$ {Y_{-3}^{(3)}} $	$ {Y_{-2}^{(3)}} $	$ {Y_{-1}^{(3)}} $	$ {Y_{0}^{(3)}} $	$ {Y_{1}^{(3)}} $	$ {Y_{2}^{(3)}} $	$ {Y_{3}^{(3)}} $
$ {Y_{-1}^{(1)}} $	$ -\frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} $	$ (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right] $	$ \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} $	$ (2 \text{Ma2u1}+\text{Meu2}) \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right] $	$ -\frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} $	$ \frac{(-1)^{3/4} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} $	$ \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} $
$ {Y_{0}^{(1)}} $	$ \left(-\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) $	$ -\frac{i \text{Ma2u1}}{\sqrt{6}} $	$ (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right] $	$ -\frac{2 \text{Ma2u1}}{3 \sqrt{5}}+\text{Meu1}-\frac{\text{Meu2}}{3 \sqrt{5}} $	$ (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right] $	$ \frac{i \text{Ma2u1}}{\sqrt{6}} $	$ \left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) $
$ {Y_{1}^{(1)}} $	$ \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} $	$ \frac{\sqrt[4]{-1} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} $	$ \frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} $	$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) (2 \text{Ma2u1}+\text{Meu2})}{\sqrt{10}} $	$ \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} $	$ (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right] $	$ \frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} $

The Hamiltonian on a basis of symmetric functions

$ $	$ f_{\text{xyz}} $	$ f_{x^3+y^3+z^3} $	$ f_{x^3-y^3} $	$ f_{2z^3-x^3-y^3} $	$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $	$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $	$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $
$ p_{x+y+z} $	$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(2-2 i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(-1+i)\right)\right) $	$ \left(-\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(-15 \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+15 i \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]-60 \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+(29+29 i) \sqrt{5}\right)+\text{Meu2} \left(-45 \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+45 i \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]-30 \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+(7+7 i) \sqrt{5}\right)-(90+90 i) \text{Meu1}\right) $	$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)}{\sqrt{2}} $	$ \frac{\text{Ma2u1} \left(-(3-3 i) \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+(3+3 i) \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+(24-24 i) \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+2 \sqrt{5}\right)+(1+i) \text{Meu2} \left(9 i \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+9 \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]-12 i \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+(-1+i) \sqrt{5}\right)}{36 \sqrt{2}} $	$ \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)\right) $	$ -\frac{\left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(4 \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+4 i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)+\text{Meu2} \left(4 \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+4 i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)\right)}{\sqrt{2}} $	$ \frac{\left(\frac{1}{12}+\frac{i}{12}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(i \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+(-1+i)\right)}{\sqrt{2}} $
$ p_{x-y} $	$ \frac{\left(\frac{1}{2}+\frac{i}{2}\right) (\text{Ma2u1}+\text{Meu2}) \left(\text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]\right)}{\sqrt{2}} $	$ \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+i \sqrt{10}\right)}{30 \sqrt{3}} $	$ \text{Meu1} $	$ \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+i \sqrt{10}\right)}{15 \sqrt{6}} $	$ \frac{\left(\frac{1}{6}+\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(-1+i)\right)}{\sqrt{2}} $	$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(-1+i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(2-2 i)\right)\right) $	$ 0 $
$ p_{3z-r} $	$ \frac{\left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+(1+i)\right)}{\sqrt{2}} $	$ \left(\frac{1}{180}-\frac{i}{180}\right) \left(\text{Ma2u1} \left(-4 \text{Root}\left[\text{$\#$1}^4+25\$	$,1\right]+15 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]-15 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+(1+i) \sqrt{10}\right)+\text{Meu2} \left(-2 \text{Root}\left[\text{$\#$1}^4+25\$	$,1\right]+45 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]-45 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+(8+8 i) \sqrt{10}\right)\right) $	$ \left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right) $	$ \frac{\left(\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(15 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+15 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+120 i \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+(-7+7 i) \sqrt{10}\right)+\text{Meu2} \left(45 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+45 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+60 i \text{Root}\left[2025 \text{$\#$1}^4+1\$	$,1\right]+(-11+11 i) \sqrt{10}\right)+(90-90 i) \sqrt{2} \text{Meu1}\right)}{\sqrt{2}} $	$ \frac{\left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(\text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)+\text{Meu2} \left(\text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)\right)}{\sqrt{2}} $	$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$	$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$	$,3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]\right)\right)\right) $	$ \frac{1}{12} \left(\text{Ma2u1} \left(-(1-i) \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+(1+i) \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]-2\right)+(3+3 i) \text{Meu2} \left(i \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,1\right]+\sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$	$,3\right]+(1-i)\right)\right) $

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

Table of Contents

Orientation 111

Symmetry Operations

Different Settings

Character Table

Product Table

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Expansion

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Quanty

One particle coupling on a basis of spherical harmonics

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

One particle coupling on a basis of symmetry adapted functions

Coupling for a single shell

Potential for s orbitals

Potential for p orbitals

Potential for d orbitals

Potential for f orbitals

Coupling between two shells

Potential for s-d orbital mixing

Potential for p-f orbital mixing

Table of several point groups