Table of Contents

Orientation X

Symmetry Operations

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

Operator Orientation
$\text{E}$ $\{0,0,0\}$ ,
$\sigma _h$ $\{1,0,0\}$ ,

Different Settings

Character Table

$ $ $ \text{E} \,{\text{(1)}} $ $ \sigma_h \,{\text{(1)}} $
$ \text{A'} $ $ 1 $ $ 1 $
$ \text{A''} $ $ 1 $ $ -1 $

Product Table

$ $ $ \text{A'} $ $ \text{A''} $
$ \text{A'} $ $ \text{A'} $ $ \text{A''} $
$ \text{A''} $ $ \text{A''} $ $ \text{A'} $

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Any potential (function) can be written as a sum over spherical harmonics. $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Cs Point group with orientation X the form of the expansion coefficients is:

Expansion

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

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

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

One particle coupling on a basis of spherical harmonics

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

we can express the operator as $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$

The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter.

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

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

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

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

One particle coupling on a basis of symmetry adapted functions

After rotation we find

$ $ $ \text{s} $ $ p_y $ $ p_z $ $ p_x $ $ d_{y^2-z^2} $ $ d_{3x^2-r^2} $ $ d_{\text{xz}} $ $ d_{\text{xy}} $ $ d_{\text{yz}} $ $ f_{\text{xyz}} $ $ f_{y\left(5y^2-r^2\right)} $ $ f_{z\left(5z^2-r^2\right)} $ $ f_{x\left(5x^2-r^2\right)} $ $ f_{y\left(z^2-x^2\right)} $ $ f_{z\left(x^2-y^2\right)} $ $ f_{x\left(y^2-z^2\right)} $
$ \text{s} $$ \text{Ass}(0,0) $$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$$\color{darkred}{ 0 }$$ -\frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} $$ \sqrt{\frac{3}{10}} \text{Asd}(2,2)-\frac{\text{Asd}(2,0)}{2 \sqrt{5}} $$ 0 $$ 0 $$ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $$\color{darkred}{ 0 }$$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$$\color{darkred}{ 0 }$$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$$\color{darkred}{ 0 }$
$ p_y $$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $$ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $$ 0 $$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)-\frac{3}{14} \text{Bpd}(3,3) }$$\color{darkred}{ -\sqrt{\frac{2}{15}} \text{Bpd}(1,1)-\frac{3 \text{Bpd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Bpd}(3,3) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$$ 0 $$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $$ \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) $$ 0 $$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $$ 0 $
$ p_z $$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$$ \frac{1}{5} \sqrt{6} \text{Bpp}(2,1) $$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $$ 0 $$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }$$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)+\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$$ 0 $$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) $$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $$ 0 $$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) $$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $$ 0 $
$ p_x $$\color{darkred}{ 0 }$$ 0 $$ 0 $$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) $$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$$\color{darkred}{ 0 }$$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $$ 0 $$ 0 $$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $$ 0 $$ 0 $$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $
$ d_{y^2-z^2} $$ -\frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} $$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)-\frac{3}{14} \text{Bpd}(3,3) }$$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }$$\color{darkred}{ 0 }$$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{19}{84} \text{Add}(4,0)+\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{6} \sqrt{\frac{5}{14}} \text{Add}(4,4) $$ \frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)+\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)-\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) $$ 0 $$ 0 $$ -\frac{1}{6} \sqrt{5} \text{Bdd}(4,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $$\color{darkred}{ 0 }$$\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,3)+\frac{5 \text{Bdf}(5,1)}{6 \sqrt{14}}+\frac{5}{44} \sqrt{3} \text{Bdf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$$\color{darkred}{ -\frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)-\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }$$\color{darkred}{ 0 }$$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{17}{22} \sqrt{\frac{5}{42}} \text{Bdf}(5,1)+\frac{1}{132} \sqrt{5} \text{Bdf}(5,3)+\frac{5}{44} \text{Bdf}(5,5) }$$\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$$\color{darkred}{ 0 }$
$ d_{3x^2-r^2} $$ \sqrt{\frac{3}{10}} \text{Asd}(2,2)-\frac{\text{Asd}(2,0)}{2 \sqrt{5}} $$\color{darkred}{ -\sqrt{\frac{2}{15}} \text{Bpd}(1,1)-\frac{3 \text{Bpd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Bpd}(3,3) }$$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)+\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }$$\color{darkred}{ 0 }$$ \frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)+\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)-\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) $$ \text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{3}{28} \text{Add}(4,0)-\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{14}} \text{Add}(4,4) $$ 0 $$ 0 $$ -\frac{2}{7} \sqrt{2} \text{Bdd}(2,1)-\frac{1}{14} \sqrt{\frac{5}{3}} \text{Bdd}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdd}(4,3) $$\color{darkred}{ 0 }$$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{7}{15}} \text{Bdf}(3,1)-\frac{\text{Bdf}(3,3)}{6 \sqrt{7}}+\frac{5 \text{Bdf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{5} \text{Bdf}(5,5) }$$\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}-\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)+\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{3}{14}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)-\frac{3}{22} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)+\frac{7}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{3} \text{Bdf}(5,5) }$$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{\sqrt{21}}+\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }$$\color{darkred}{ 0 }$
$ d_{\text{xz}} $$ 0 $$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$$ 0 $$ 0 $$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $$ \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $$ 0 $$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$
$ d_{\text{xy}} $$ 0 $$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$$ 0 $$ 0 $$ \frac{1}{7} \sqrt{6} \text{Bdd}(2,1)-\frac{1}{21} \sqrt{5} \text{Bdd}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $$ 0 $$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$
$ d_{\text{yz}} $$ \sqrt{\frac{2}{5}} \text{Bsd}(2,1) $$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$$\color{darkred}{ 0 }$$ -\frac{1}{6} \sqrt{5} \text{Bdd}(4,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $$ -\frac{2}{7} \sqrt{2} \text{Bdd}(2,1)-\frac{1}{14} \sqrt{\frac{5}{3}} \text{Bdd}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdd}(4,3) $$ 0 $$ 0 $$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$$\color{darkred}{ 0 }$$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$$\color{darkred}{ 0 }$
$ f_{\text{xyz}} $$\color{darkred}{ 0 }$$ 0 $$ 0 $$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)-\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$$\color{darkred}{ 0 }$$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $$ 0 $$ 0 $$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $$ 0 $$ 0 $$ -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $
$ f_{y\left(5y^2-r^2\right)} $$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1)-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Bpf}(4,3) $$ 0 $$\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,3)+\frac{5 \text{Bdf}(5,1)}{6 \sqrt{14}}+\frac{5}{44} \sqrt{3} \text{Bdf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{7}{15}} \text{Bdf}(3,1)-\frac{\text{Bdf}(3,3)}{6 \sqrt{7}}+\frac{5 \text{Bdf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{5} \text{Bdf}(5,5) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$$ 0 $$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $$ -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) $$ 0 $$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $$ -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $$ 0 $
$ f_{z\left(5z^2-r^2\right)} $$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$$ \frac{2}{3} \sqrt{\frac{5}{21}} \text{Bpf}(4,1)-\frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,1) $$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $$ 0 $$\color{darkred}{ -\frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)-\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }$$\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}-\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)+\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$$ 0 $$ -\frac{\text{Bff}(2,1)}{5 \sqrt{6}}-\frac{1}{22} \sqrt{5} \text{Bff}(4,1)+\frac{1}{22} \sqrt{35} \text{Bff}(4,3)-\frac{25}{143} \sqrt{\frac{7}{6}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,3) $$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $$ 0 $$ \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) $$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $
$ f_{x\left(5x^2-r^2\right)} $$\color{darkred}{ 0 }$$ 0 $$ 0 $$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$$\color{darkred}{ 0 }$$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,1)+\frac{\text{Bff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)+\frac{5}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)+\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $$ 0 $$ 0 $$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $$ 0 $$ 0 $$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $
$ f_{y\left(z^2-x^2\right)} $$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $$ \sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{5 \text{Bpf}(4,1)}{6 \sqrt{7}}-\frac{1}{6} \text{Bpf}(4,3) $$ 0 $$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{17}{22} \sqrt{\frac{5}{42}} \text{Bdf}(5,1)+\frac{1}{132} \sqrt{5} \text{Bdf}(5,3)+\frac{5}{44} \text{Bdf}(5,5) }$$\color{darkred}{ -\sqrt{\frac{3}{14}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)-\frac{3}{22} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)+\frac{7}{44} \sqrt{\frac{5}{3}} \text{Bdf}(5,3)-\frac{5}{44} \sqrt{3} \text{Bdf}(5,5) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$$ 0 $$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $$ \frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Bff}(4,1)}{22 \sqrt{3}}+\frac{1}{22} \sqrt{21} \text{Bff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Bff}(6,1)-\frac{5}{143} \sqrt{7} \text{Bff}(6,3) $$ 0 $$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $$ -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $$ 0 $
$ f_{z\left(x^2-y^2\right)} $$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,1)+\frac{\text{Bpf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Bpf}(4,3) $$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $$ 0 $$\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{\sqrt{21}}+\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }$$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$$ 0 $$ -\frac{\text{Bff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Bff}(4,1)-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Bff}(4,3)-\frac{5}{429} \sqrt{70} \text{Bff}(6,1)-\frac{15}{286} \sqrt{7} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Bff}(6,5) $$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $$ -\frac{\text{Bff}(2,1)}{\sqrt{6}}-\frac{1}{66} \sqrt{5} \text{Bff}(4,1)+\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,1)+\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $$ 0 $
$ f_{x\left(y^2-z^2\right)} $$\color{darkred}{ 0 }$$ 0 $$ 0 $$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $$\color{darkred}{ 0 }$$\color{darkred}{ 0 }$$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$$\color{darkred}{ 0 }$$ -\frac{7}{66} \sqrt{5} \text{Bff}(4,1)-\frac{1}{66} \sqrt{35} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Bff}(6,1)-\frac{5}{286} \sqrt{105} \text{Bff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Bff}(6,5) $$ 0 $$ 0 $$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $$ 0 $$ 0 $$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $

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

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_X.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Eap, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_X.Quanty
Akm = {{0, 0, Eap} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ \text{s} $
$ \text{s} $$ \text{Eap} $

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

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

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Eapp}+\text{Eapy}+\text{Eapz}) & k=0\land m=0 \\ 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ \frac{5 (\text{Eapp}-\text{Eapy})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\ \frac{5 i \text{Mapyz}}{\sqrt{6}} & k=2\land (m=-1\lor m=1) \\ -\frac{5}{6} (\text{Eapp}+\text{Eapy}-2 \text{Eapz}) & \text{True} \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_X.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eapp + Eapy + Eapz)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {(5*(Eapp - Eapy))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}, {((5*I)*Mapyz)/Sqrt[6], k == 2 && (m == -1 || m == 1)}}, (-5*(Eapp + Eapy - 2*Eapz))/6]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_X.Quanty
Akm = {{0, 0, (1/3)*(Eapp + Eapy + Eapz)} , 
       {2, 0, (-5/6)*(Eapp + Eapy + (-2)*(Eapz))} , 
       {2,-1, (5*I)*((1/(sqrt(6)))*(Mapyz))} , 
       {2, 1, (5*I)*((1/(sqrt(6)))*(Mapyz))} , 
       {2,-2, (5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapy)))} , 
       {2, 2, (5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapy)))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{-1}^{(1)}} $ $ {Y_{0}^{(1)}} $ $ {Y_{1}^{(1)}} $
$ {Y_{-1}^{(1)}} $$ \frac{\text{Eapp}+\text{Eapy}}{2} $$ \frac{i \text{Mapyz}}{\sqrt{2}} $$ \frac{\text{Eapy}-\text{Eapp}}{2} $
$ {Y_{0}^{(1)}} $$ -\frac{i \text{Mapyz}}{\sqrt{2}} $$ \text{Eapz} $$ -\frac{i \text{Mapyz}}{\sqrt{2}} $
$ {Y_{1}^{(1)}} $$ \frac{\text{Eapy}-\text{Eapp}}{2} $$ \frac{i \text{Mapyz}}{\sqrt{2}} $$ \frac{\text{Eapp}+\text{Eapy}}{2} $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ p_y $ $ p_z $ $ p_x $
$ p_y $$ \text{Eapy} $$ \text{Mapyz} $$ 0 $
$ p_z $$ \text{Mapyz} $$ \text{Eapz} $$ 0 $
$ p_x $$ 0 $$ 0 $$ \text{Eapp} $

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

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

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Eappxy}+\text{Eappxz}+\text{Eapx2}+\text{Eapy2z2}+\text{Eapyz}) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ \frac{1}{4} \left(\sqrt{6} \text{Eappxz}+\sqrt{6} \text{Eapx2}-\sqrt{6} \text{Eapy2z2}-\sqrt{6} \text{Eapyz}+2 \sqrt{2} \text{Mapy2z2x2}\right) & k=2\land (m=-2\lor m=2) \\ \frac{1}{2} i \left(\sqrt{6} \text{Mappxzxy}-2 \sqrt{2} \text{Mapx2yz}\right) & k=2\land (m=-1\lor m=1) \\ \frac{1}{2} \left(-2 \text{Eappxy}+\text{Eappxz}-\text{Eapx2}+\text{Eapy2z2}+\text{Eapyz}+2 \sqrt{3} \text{Mapy2z2x2}\right) & k=2\land m=0 \\ -\frac{3}{8} \sqrt{\frac{7}{10}} \left(4 \text{Eappxy}-3 \text{Eapx2}-\text{Eapy2z2}+2 \sqrt{3} \text{Mapy2z2x2}\right) & k=4\land (m=-4\lor m=4) \\ \frac{3}{4} i \sqrt{\frac{7}{5}} \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) & k=4\land (m=-3\lor m=3) \\ \frac{3 \left(4 \text{Eappxz}-3 \text{Eapx2}+3 \text{Eapy2z2}-4 \text{Eapyz}-2 \sqrt{3} \text{Mapy2z2x2}\right)}{4 \sqrt{10}} & k=4\land (m=-2\lor m=2) \\ -\frac{3 i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}+7 \text{Mapy2z2yz}\right)}{4 \sqrt{5}} & k=4\land (m=-1\lor m=1) \\ \frac{3}{40} \left(4 \text{Eappxy}-16 \text{Eappxz}+9 \text{Eapx2}+19 \text{Eapy2z2}-16 \text{Eapyz}+10 \sqrt{3} \text{Mapy2z2x2}\right) & \text{True} \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_X.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eappxy + Eappxz + Eapx2 + Eapy2z2 + Eapyz)/5, k == 0 && m == 0}, {0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(Sqrt[6]*Eappxz + Sqrt[6]*Eapx2 - Sqrt[6]*Eapy2z2 - Sqrt[6]*Eapyz + 2*Sqrt[2]*Mapy2z2x2)/4, k == 2 && (m == -2 || m == 2)}, {(I/2)*(Sqrt[6]*Mappxzxy - 2*Sqrt[2]*Mapx2yz), k == 2 && (m == -1 || m == 1)}, {(-2*Eappxy + Eappxz - Eapx2 + Eapy2z2 + Eapyz + 2*Sqrt[3]*Mapy2z2x2)/2, k == 2 && m == 0}, {(-3*Sqrt[7/10]*(4*Eappxy - 3*Eapx2 - Eapy2z2 + 2*Sqrt[3]*Mapy2z2x2))/8, k == 4 && (m == -4 || m == 4)}, {((3*I)/4)*Sqrt[7/5]*(2*Mappxzxy + Sqrt[3]*Mapx2yz - Mapy2z2yz), k == 4 && (m == -3 || m == 3)}, {(3*(4*Eappxz - 3*Eapx2 + 3*Eapy2z2 - 4*Eapyz - 2*Sqrt[3]*Mapy2z2x2))/(4*Sqrt[10]), k == 4 && (m == -2 || m == 2)}, {(((-3*I)/4)*(2*Mappxzxy + Sqrt[3]*Mapx2yz + 7*Mapy2z2yz))/Sqrt[5], k == 4 && (m == -1 || m == 1)}}, (3*(4*Eappxy - 16*Eappxz + 9*Eapx2 + 19*Eapy2z2 - 16*Eapyz + 10*Sqrt[3]*Mapy2z2x2))/40]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_X.Quanty
Akm = {{0, 0, (1/5)*(Eappxy + Eappxz + Eapx2 + Eapy2z2 + Eapyz)} , 
       {2, 0, (1/2)*((-2)*(Eappxy) + Eappxz + (-1)*(Eapx2) + Eapy2z2 + Eapyz + (2)*((sqrt(3))*(Mapy2z2x2)))} , 
       {2,-1, (1/2*I)*((sqrt(6))*(Mappxzxy) + (-2)*((sqrt(2))*(Mapx2yz)))} , 
       {2, 1, (1/2*I)*((sqrt(6))*(Mappxzxy) + (-2)*((sqrt(2))*(Mapx2yz)))} , 
       {2,-2, (1/4)*((sqrt(6))*(Eappxz) + (sqrt(6))*(Eapx2) + (-1)*((sqrt(6))*(Eapy2z2)) + (-1)*((sqrt(6))*(Eapyz)) + (2)*((sqrt(2))*(Mapy2z2x2)))} , 
       {2, 2, (1/4)*((sqrt(6))*(Eappxz) + (sqrt(6))*(Eapx2) + (-1)*((sqrt(6))*(Eapy2z2)) + (-1)*((sqrt(6))*(Eapyz)) + (2)*((sqrt(2))*(Mapy2z2x2)))} , 
       {4, 0, (3/40)*((4)*(Eappxy) + (-16)*(Eappxz) + (9)*(Eapx2) + (19)*(Eapy2z2) + (-16)*(Eapyz) + (10)*((sqrt(3))*(Mapy2z2x2)))} , 
       {4,-1, (-3/4*I)*((1/(sqrt(5)))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (7)*(Mapy2z2yz)))} , 
       {4, 1, (-3/4*I)*((1/(sqrt(5)))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (7)*(Mapy2z2yz)))} , 
       {4,-2, (3/4)*((1/(sqrt(10)))*((4)*(Eappxz) + (-3)*(Eapx2) + (3)*(Eapy2z2) + (-4)*(Eapyz) + (-2)*((sqrt(3))*(Mapy2z2x2))))} , 
       {4, 2, (3/4)*((1/(sqrt(10)))*((4)*(Eappxz) + (-3)*(Eapx2) + (3)*(Eapy2z2) + (-4)*(Eapyz) + (-2)*((sqrt(3))*(Mapy2z2x2))))} , 
       {4,-3, (3/4*I)*((sqrt(7/5))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (-1)*(Mapy2z2yz)))} , 
       {4, 3, (3/4*I)*((sqrt(7/5))*((2)*(Mappxzxy) + (sqrt(3))*(Mapx2yz) + (-1)*(Mapy2z2yz)))} , 
       {4,-4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapx2) + (-1)*(Eapy2z2) + (2)*((sqrt(3))*(Mapy2z2x2))))} , 
       {4, 4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapx2) + (-1)*(Eapy2z2) + (2)*((sqrt(3))*(Mapy2z2x2))))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{-2}^{(2)}} $ $ {Y_{-1}^{(2)}} $ $ {Y_{0}^{(2)}} $ $ {Y_{1}^{(2)}} $ $ {Y_{2}^{(2)}} $
$ {Y_{-2}^{(2)}} $$ \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $$ \frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $$ -\frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $$ \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $
$ {Y_{-1}^{(2)}} $$ -\frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $$ \frac{\text{Eappxz}+\text{Eapyz}}{2} $$ -\frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $$ \frac{\text{Eapyz}-\text{Eappxz}}{2} $$ \frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $
$ {Y_{0}^{(2)}} $$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $$ \frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $$ \frac{1}{4} \left(\text{Eapx2}+3 \text{Eapy2z2}+2 \sqrt{3} \text{Mapy2z2x2}\right) $$ \frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $
$ {Y_{1}^{(2)}} $$ \frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $$ \frac{\text{Eapyz}-\text{Eappxz}}{2} $$ -\frac{i \left(\text{Mapx2yz}+\sqrt{3} \text{Mapy2z2yz}\right)}{2 \sqrt{2}} $$ \frac{\text{Eappxz}+\text{Eapyz}}{2} $$ -\frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $
$ {Y_{2}^{(2)}} $$ \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $$ -\frac{1}{4} i \left(2 \text{Mappxzxy}+\sqrt{3} \text{Mapx2yz}-\text{Mapy2z2yz}\right) $$ \frac{1}{8} \left(-\sqrt{6} \text{Eapx2}+\sqrt{6} \text{Eapy2z2}-2 \sqrt{2} \text{Mapy2z2x2}\right) $$ \frac{1}{4} i \left(2 \text{Mappxzxy}-\sqrt{3} \text{Mapx2yz}+\text{Mapy2z2yz}\right) $$ \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapx2}+\text{Eapy2z2}-2 \sqrt{3} \text{Mapy2z2x2}\right) $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ d_{y^2-z^2} $ $ d_{3x^2-r^2} $ $ d_{\text{xz}} $ $ d_{\text{xy}} $ $ d_{\text{yz}} $
$ d_{y^2-z^2} $$ \text{Eapy2z2} $$ \text{Mapy2z2x2} $$ 0 $$ 0 $$ \text{Mapy2z2yz} $
$ d_{3x^2-r^2} $$ \text{Mapy2z2x2} $$ \text{Eapx2} $$ 0 $$ 0 $$ \text{Mapx2yz} $
$ d_{\text{xz}} $$ 0 $$ 0 $$ \text{Eappxz} $$ \text{Mappxzxy} $$ 0 $
$ d_{\text{xy}} $$ 0 $$ 0 $$ \text{Mappxzxy} $$ \text{Eappxy} $$ 0 $
$ d_{\text{yz}} $$ \text{Mapy2z2yz} $$ \text{Mapx2yz} $$ 0 $$ 0 $$ \text{Eapyz} $

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

$$\text{Eapy2z2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{15}{\pi }} \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{8} \sqrt{\frac{15}{\pi }} \left(-x^2+y^2-3 z^2+1\right)$$
$$\text{Eapx2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{5}{\pi }} \left(-6 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{8} \sqrt{\frac{5}{\pi }} \left(3 x^2-3 y^2-3 z^2+1\right)$$
$$\text{Eappxz}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$
$$\text{Eappxy}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$
$$\text{Eapyz}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Eappx3}+\text{Eappxy2z2}+\text{Eappxyz}+\text{Eapy3}+\text{Eapyz2x2}+\text{Eapz3}+\text{Eapzx2y2}) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4\land k\neq 6)\lor (k\neq 4\land k\neq 6\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (k\neq 6\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4)\lor (m\neq -6\land m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5\land m\neq 6) \\ \frac{5}{28} \left(\sqrt{6} \text{Eappx3}-\sqrt{6} \text{Eapy3}+\sqrt{10} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2}-2 \text{Mapz3zx2y2})\right) & k=2\land (m=-2\lor m=2) \\ -\frac{5}{56} i \left(4 \sqrt{10} \text{Mappxyzx3}+\sqrt{6} \text{Mapy3z3}+\sqrt{10} \text{Mapy3zx2y2}+5 \sqrt{6} \text{Mapyz2x2zx2y2}-\sqrt{10} \text{Mapz3yz2x2}\right) & k=2\land (m=-1\lor m=1) \\ -\frac{5}{14} \left(\text{Eappx3}+\text{Eapy3}-2 \text{Eapz3}-\sqrt{15} \text{Mappx3xy2z2}+\sqrt{15} \text{Mapy3yz2x2}\right) & k=2\land m=0 \\ \frac{3 \left(3 \sqrt{5} \text{Eappx3}-3 \sqrt{5} \text{Eappxy2z2}-4 \sqrt{5} \text{Eappxyz}+3 \sqrt{5} \text{Eapy3}-3 \sqrt{5} \text{Eapyz2x2}+4 \sqrt{5} \text{Eapzx2y2}+2 \sqrt{3} \text{Mappx3xy2z2}-2 \sqrt{3} \text{Mapy3yz2x2}\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\ -\frac{3 i \left(\sqrt{3} \text{Mappxyzx3}+\sqrt{5} \text{Mappxyzxy2z2}-3 \sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapy3zx2y2}-\sqrt{5} \text{Mapyz2x2zx2y2}-3 \sqrt{3} \text{Mapz3yz2x2}\right)}{4 \sqrt{7}} & k=4\land (m=-3\lor m=3) \\ -\frac{3}{56} \left(3 \sqrt{10} \text{Eappx3}-7 \sqrt{10} \text{Eappxy2z2}-3 \sqrt{10} \text{Eapy3}+7 \sqrt{10} \text{Eapyz2x2}-2 \sqrt{6} \text{Mappx3xy2z2}-2 \sqrt{6} \text{Mapy3yz2x2}+4 \sqrt{6} \text{Mapz3zx2y2}\right) & k=4\land (m=-2\lor m=2) \\ \frac{3}{28} i \left(\sqrt{3} \text{Mappxyzx3}-7 \sqrt{5} \text{Mappxyzxy2z2}-3 \sqrt{5} \text{Mapy3z3}+9 \sqrt{3} \text{Mapy3zx2y2}-\sqrt{5} \text{Mapyz2x2zx2y2}+5 \sqrt{3} \text{Mapz3yz2x2}\right) & k=4\land (m=-1\lor m=1) \\ \frac{3}{56} \left(9 \text{Eappx3}+7 \text{Eappxy2z2}-28 \text{Eappxyz}+9 \text{Eapy3}+7 \text{Eapyz2x2}+24 \text{Eapz3}-28 \text{Eapzx2y2}-2 \sqrt{15} \text{Mappx3xy2z2}+2 \sqrt{15} \text{Mapy3yz2x2}\right) & k=4\land m=0 \\ \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappx3}+3 \sqrt{3} \text{Eappxy2z2}-5 \sqrt{3} \text{Eapy3}-3 \sqrt{3} \text{Eapyz2x2}-6 \sqrt{5} \text{Mappx3xy2z2}-6 \sqrt{5} \text{Mapy3yz2x2}\right) & k=6\land (m=-6\lor m=6) \\ \frac{13}{40} i \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right) & k=6\land (m=-5\lor m=5) \\ -\frac{13 \left(15 \text{Eappx3}-15 \text{Eappxy2z2}+24 \text{Eappxyz}+15 \text{Eapy3}-15 \text{Eapyz2x2}-24 \text{Eapzx2y2}+2 \sqrt{15} \text{Mappx3xy2z2}-2 \sqrt{15} \text{Mapy3yz2x2}\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ -\frac{13 i \left(9 \text{Mappxyzx3}+3 \sqrt{15} \text{Mappxyzxy2z2}+2 \sqrt{15} \text{Mapy3z3}+9 \text{Mapy3zx2y2}-3 \sqrt{15} \text{Mapyz2x2zx2y2}+6 \text{Mapz3yz2x2}\right)}{40 \sqrt{7}} & k=6\land (m=-3\lor m=3) \\ \frac{13 \left(5 \sqrt{15} \text{Eappx3}+3 \sqrt{15} \text{Eappxy2z2}-5 \sqrt{15} \text{Eapy3}-3 \sqrt{15} \text{Eapyz2x2}+34 \text{Mappx3xy2z2}+34 \text{Mapy3yz2x2}+64 \text{Mapz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\ \frac{13}{280} i \left(\sqrt{70} \text{Mappxyzx3}+3 \sqrt{42} \text{Mappxyzxy2z2}-5 \sqrt{42} \text{Mapy3z3}-2 \sqrt{70} \text{Mapy3zx2y2}+2 \sqrt{42} \text{Mapyz2x2zx2y2}+5 \sqrt{70} \text{Mapz3yz2x2}\right) & k=6\land (m=-1\lor m=1) \\ -\frac{13}{560} \left(25 \text{Eappx3}+39 \text{Eappxy2z2}-24 \text{Eappxyz}+25 \text{Eapy3}+39 \text{Eapyz2x2}-80 \text{Eapz3}-24 \text{Eapzx2y2}+14 \sqrt{15} \text{Mappx3xy2z2}-14 \sqrt{15} \text{Mapy3yz2x2}\right) & \text{True} \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Cs_X.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eappx3 + Eappxy2z2 + Eappxyz + Eapy3 + Eapyz2x2 + Eapz3 + Eapzx2y2)/7, k == 0 && m == 0}, {0, (k != 2 && k != 4 && k != 6) || (k != 4 && k != 6 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (k != 6 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4) || (m != -6 && m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5 && m != 6)}, {(5*(Sqrt[6]*Eappx3 - Sqrt[6]*Eapy3 + Sqrt[10]*(Mappx3xy2z2 + Mapy3yz2x2 - 2*Mapz3zx2y2)))/28, k == 2 && (m == -2 || m == 2)}, {((-5*I)/56)*(4*Sqrt[10]*Mappxyzx3 + Sqrt[6]*Mapy3z3 + Sqrt[10]*Mapy3zx2y2 + 5*Sqrt[6]*Mapyz2x2zx2y2 - Sqrt[10]*Mapz3yz2x2), k == 2 && (m == -1 || m == 1)}, {(-5*(Eappx3 + Eapy3 - 2*Eapz3 - Sqrt[15]*Mappx3xy2z2 + Sqrt[15]*Mapy3yz2x2))/14, k == 2 && m == 0}, {(3*(3*Sqrt[5]*Eappx3 - 3*Sqrt[5]*Eappxy2z2 - 4*Sqrt[5]*Eappxyz + 3*Sqrt[5]*Eapy3 - 3*Sqrt[5]*Eapyz2x2 + 4*Sqrt[5]*Eapzx2y2 + 2*Sqrt[3]*Mappx3xy2z2 - 2*Sqrt[3]*Mapy3yz2x2))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(((-3*I)/4)*(Sqrt[3]*Mappxyzx3 + Sqrt[5]*Mappxyzxy2z2 - 3*Sqrt[5]*Mapy3z3 + Sqrt[3]*Mapy3zx2y2 - Sqrt[5]*Mapyz2x2zx2y2 - 3*Sqrt[3]*Mapz3yz2x2))/Sqrt[7], k == 4 && (m == -3 || m == 3)}, {(-3*(3*Sqrt[10]*Eappx3 - 7*Sqrt[10]*Eappxy2z2 - 3*Sqrt[10]*Eapy3 + 7*Sqrt[10]*Eapyz2x2 - 2*Sqrt[6]*Mappx3xy2z2 - 2*Sqrt[6]*Mapy3yz2x2 + 4*Sqrt[6]*Mapz3zx2y2))/56, k == 4 && (m == -2 || m == 2)}, {((3*I)/28)*(Sqrt[3]*Mappxyzx3 - 7*Sqrt[5]*Mappxyzxy2z2 - 3*Sqrt[5]*Mapy3z3 + 9*Sqrt[3]*Mapy3zx2y2 - Sqrt[5]*Mapyz2x2zx2y2 + 5*Sqrt[3]*Mapz3yz2x2), k == 4 && (m == -1 || m == 1)}, {(3*(9*Eappx3 + 7*Eappxy2z2 - 28*Eappxyz + 9*Eapy3 + 7*Eapyz2x2 + 24*Eapz3 - 28*Eapzx2y2 - 2*Sqrt[15]*Mappx3xy2z2 + 2*Sqrt[15]*Mapy3yz2x2))/56, k == 4 && m == 0}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eappx3 + 3*Sqrt[3]*Eappxy2z2 - 5*Sqrt[3]*Eapy3 - 3*Sqrt[3]*Eapyz2x2 - 6*Sqrt[5]*Mappx3xy2z2 - 6*Sqrt[5]*Mapy3yz2x2))/160, k == 6 && (m == -6 || m == 6)}, {((13*I)/40)*Sqrt[11/7]*(Sqrt[15]*Mappxyzx3 - 3*Mappxyzxy2z2 - Sqrt[15]*Mapy3zx2y2 - 3*Mapyz2x2zx2y2), k == 6 && (m == -5 || m == 5)}, {(-13*(15*Eappx3 - 15*Eappxy2z2 + 24*Eappxyz + 15*Eapy3 - 15*Eapyz2x2 - 24*Eapzx2y2 + 2*Sqrt[15]*Mappx3xy2z2 - 2*Sqrt[15]*Mapy3yz2x2))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(((-13*I)/40)*(9*Mappxyzx3 + 3*Sqrt[15]*Mappxyzxy2z2 + 2*Sqrt[15]*Mapy3z3 + 9*Mapy3zx2y2 - 3*Sqrt[15]*Mapyz2x2zx2y2 + 6*Mapz3yz2x2))/Sqrt[7], k == 6 && (m == -3 || m == 3)}, {(13*(5*Sqrt[15]*Eappx3 + 3*Sqrt[15]*Eappxy2z2 - 5*Sqrt[15]*Eapy3 - 3*Sqrt[15]*Eapyz2x2 + 34*Mappx3xy2z2 + 34*Mapy3yz2x2 + 64*Mapz3zx2y2))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {((13*I)/280)*(Sqrt[70]*Mappxyzx3 + 3*Sqrt[42]*Mappxyzxy2z2 - 5*Sqrt[42]*Mapy3z3 - 2*Sqrt[70]*Mapy3zx2y2 + 2*Sqrt[42]*Mapyz2x2zx2y2 + 5*Sqrt[70]*Mapz3yz2x2), k == 6 && (m == -1 || m == 1)}}, (-13*(25*Eappx3 + 39*Eappxy2z2 - 24*Eappxyz + 25*Eapy3 + 39*Eapyz2x2 - 80*Eapz3 - 24*Eapzx2y2 + 14*Sqrt[15]*Mappx3xy2z2 - 14*Sqrt[15]*Mapy3yz2x2))/560]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_X.Quanty
Akm = {{0, 0, (1/7)*(Eappx3 + Eappxy2z2 + Eappxyz + Eapy3 + Eapyz2x2 + Eapz3 + Eapzx2y2)} , 
       {2, 0, (-5/14)*(Eappx3 + Eapy3 + (-2)*(Eapz3) + (-1)*((sqrt(15))*(Mappx3xy2z2)) + (sqrt(15))*(Mapy3yz2x2))} , 
       {2,-1, (-5/56*I)*((4)*((sqrt(10))*(Mappxyzx3)) + (sqrt(6))*(Mapy3z3) + (sqrt(10))*(Mapy3zx2y2) + (5)*((sqrt(6))*(Mapyz2x2zx2y2)) + (-1)*((sqrt(10))*(Mapz3yz2x2)))} , 
       {2, 1, (-5/56*I)*((4)*((sqrt(10))*(Mappxyzx3)) + (sqrt(6))*(Mapy3z3) + (sqrt(10))*(Mapy3zx2y2) + (5)*((sqrt(6))*(Mapyz2x2zx2y2)) + (-1)*((sqrt(10))*(Mapz3yz2x2)))} , 
       {2,-2, (5/28)*((sqrt(6))*(Eappx3) + (-1)*((sqrt(6))*(Eapy3)) + (sqrt(10))*(Mappx3xy2z2 + Mapy3yz2x2 + (-2)*(Mapz3zx2y2)))} , 
       {2, 2, (5/28)*((sqrt(6))*(Eappx3) + (-1)*((sqrt(6))*(Eapy3)) + (sqrt(10))*(Mappx3xy2z2 + Mapy3yz2x2 + (-2)*(Mapz3zx2y2)))} , 
       {4, 0, (3/56)*((9)*(Eappx3) + (7)*(Eappxy2z2) + (-28)*(Eappxyz) + (9)*(Eapy3) + (7)*(Eapyz2x2) + (24)*(Eapz3) + (-28)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappx3xy2z2)) + (2)*((sqrt(15))*(Mapy3yz2x2)))} , 
       {4,-1, (3/28*I)*((sqrt(3))*(Mappxyzx3) + (-7)*((sqrt(5))*(Mappxyzxy2z2)) + (-3)*((sqrt(5))*(Mapy3z3)) + (9)*((sqrt(3))*(Mapy3zx2y2)) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (5)*((sqrt(3))*(Mapz3yz2x2)))} , 
       {4, 1, (3/28*I)*((sqrt(3))*(Mappxyzx3) + (-7)*((sqrt(5))*(Mappxyzxy2z2)) + (-3)*((sqrt(5))*(Mapy3z3)) + (9)*((sqrt(3))*(Mapy3zx2y2)) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (5)*((sqrt(3))*(Mapz3yz2x2)))} , 
       {4,-2, (-3/56)*((3)*((sqrt(10))*(Eappx3)) + (-7)*((sqrt(10))*(Eappxy2z2)) + (-3)*((sqrt(10))*(Eapy3)) + (7)*((sqrt(10))*(Eapyz2x2)) + (-2)*((sqrt(6))*(Mappx3xy2z2)) + (-2)*((sqrt(6))*(Mapy3yz2x2)) + (4)*((sqrt(6))*(Mapz3zx2y2)))} , 
       {4, 2, (-3/56)*((3)*((sqrt(10))*(Eappx3)) + (-7)*((sqrt(10))*(Eappxy2z2)) + (-3)*((sqrt(10))*(Eapy3)) + (7)*((sqrt(10))*(Eapyz2x2)) + (-2)*((sqrt(6))*(Mappx3xy2z2)) + (-2)*((sqrt(6))*(Mapy3yz2x2)) + (4)*((sqrt(6))*(Mapz3zx2y2)))} , 
       {4,-3, (-3/4*I)*((1/(sqrt(7)))*((sqrt(3))*(Mappxyzx3) + (sqrt(5))*(Mappxyzxy2z2) + (-3)*((sqrt(5))*(Mapy3z3)) + (sqrt(3))*(Mapy3zx2y2) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (-3)*((sqrt(3))*(Mapz3yz2x2))))} , 
       {4, 3, (-3/4*I)*((1/(sqrt(7)))*((sqrt(3))*(Mappxyzx3) + (sqrt(5))*(Mappxyzxy2z2) + (-3)*((sqrt(5))*(Mapy3z3)) + (sqrt(3))*(Mapy3zx2y2) + (-1)*((sqrt(5))*(Mapyz2x2zx2y2)) + (-3)*((sqrt(3))*(Mapz3yz2x2))))} , 
       {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eapy3)) + (-3)*((sqrt(5))*(Eapyz2x2)) + (4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (-2)*((sqrt(3))*(Mapy3yz2x2))))} , 
       {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eapy3)) + (-3)*((sqrt(5))*(Eapyz2x2)) + (4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (-2)*((sqrt(3))*(Mapy3yz2x2))))} , 
       {6, 0, (-13/560)*((25)*(Eappx3) + (39)*(Eappxy2z2) + (-24)*(Eappxyz) + (25)*(Eapy3) + (39)*(Eapyz2x2) + (-80)*(Eapz3) + (-24)*(Eapzx2y2) + (14)*((sqrt(15))*(Mappx3xy2z2)) + (-14)*((sqrt(15))*(Mapy3yz2x2)))} , 
       {6,-1, (13/280*I)*((sqrt(70))*(Mappxyzx3) + (3)*((sqrt(42))*(Mappxyzxy2z2)) + (-5)*((sqrt(42))*(Mapy3z3)) + (-2)*((sqrt(70))*(Mapy3zx2y2)) + (2)*((sqrt(42))*(Mapyz2x2zx2y2)) + (5)*((sqrt(70))*(Mapz3yz2x2)))} , 
       {6, 1, (13/280*I)*((sqrt(70))*(Mappxyzx3) + (3)*((sqrt(42))*(Mappxyzxy2z2)) + (-5)*((sqrt(42))*(Mapy3z3)) + (-2)*((sqrt(70))*(Mapy3zx2y2)) + (2)*((sqrt(42))*(Mapyz2x2zx2y2)) + (5)*((sqrt(70))*(Mapz3yz2x2)))} , 
       {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eapy3)) + (-3)*((sqrt(15))*(Eapyz2x2)) + (34)*(Mappx3xy2z2) + (34)*(Mapy3yz2x2) + (64)*(Mapz3zx2y2)))} , 
       {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eapy3)) + (-3)*((sqrt(15))*(Eapyz2x2)) + (34)*(Mappx3xy2z2) + (34)*(Mapy3yz2x2) + (64)*(Mapz3zx2y2)))} , 
       {6,-3, (-13/40*I)*((1/(sqrt(7)))*((9)*(Mappxyzx3) + (3)*((sqrt(15))*(Mappxyzxy2z2)) + (2)*((sqrt(15))*(Mapy3z3)) + (9)*(Mapy3zx2y2) + (-3)*((sqrt(15))*(Mapyz2x2zx2y2)) + (6)*(Mapz3yz2x2)))} , 
       {6, 3, (-13/40*I)*((1/(sqrt(7)))*((9)*(Mappxyzx3) + (3)*((sqrt(15))*(Mappxyzxy2z2)) + (2)*((sqrt(15))*(Mapy3z3)) + (9)*(Mapy3zx2y2) + (-3)*((sqrt(15))*(Mapyz2x2zx2y2)) + (6)*(Mapz3yz2x2)))} , 
       {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eapy3) + (-15)*(Eapyz2x2) + (-24)*(Eapzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (-2)*((sqrt(15))*(Mapy3yz2x2))))} , 
       {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eapy3) + (-15)*(Eapyz2x2) + (-24)*(Eapzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (-2)*((sqrt(15))*(Mapy3yz2x2))))} , 
       {6,-5, (13/40*I)*((sqrt(11/7))*((sqrt(15))*(Mappxyzx3) + (-3)*(Mappxyzxy2z2) + (-1)*((sqrt(15))*(Mapy3zx2y2)) + (-3)*(Mapyz2x2zx2y2)))} , 
       {6, 5, (13/40*I)*((sqrt(11/7))*((sqrt(15))*(Mappxyzx3) + (-3)*(Mappxyzxy2z2) + (-1)*((sqrt(15))*(Mapy3zx2y2)) + (-3)*(Mapyz2x2zx2y2)))} , 
       {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eapy3)) + (-3)*((sqrt(3))*(Eapyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (-6)*((sqrt(5))*(Mapy3yz2x2))))} , 
       {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eapy3)) + (-3)*((sqrt(3))*(Eapyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (-6)*((sqrt(5))*(Mapy3yz2x2))))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{-3}^{(3)}} $ $ {Y_{-2}^{(3)}} $ $ {Y_{-1}^{(3)}} $ $ {Y_{0}^{(3)}} $ $ {Y_{1}^{(3)}} $ $ {Y_{2}^{(3)}} $ $ {Y_{3}^{(3)}} $
$ {Y_{-3}^{(3)}} $$ \frac{1}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mapy3yz2x2}-\text{Mappx3xy2z2})\right) $$ -\frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $$ -\frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $$ \frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $$ \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $
$ {Y_{-2}^{(3)}} $$ \frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $$ \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} $$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} $$ -\frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $
$ {Y_{-1}^{(3)}} $$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mapy3yz2x2})\right) $$ -\frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $$ \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}-2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $
$ {Y_{0}^{(3)}} $$ \frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $$ \frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $$ \text{Eapz3} $$ \frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $$ \frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $
$ {Y_{1}^{(3)}} $$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}-2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $$ -\frac{1}{4} i \left(\sqrt{3} \text{Mapy3z3}-\sqrt{5} \text{Mapz3yz2x2}\right) $$ \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eapy3}+5 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mapy3yz2x2})\right) $$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $
$ {Y_{2}^{(3)}} $$ -\frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $$ \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} $$ \frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}+\sqrt{6} \text{Mapy3zx2y2}-\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $$ -\frac{1}{8} i \left(\sqrt{6} \text{Mappxyzx3}+\sqrt{10} \text{Mappxyzxy2z2}-\sqrt{6} \text{Mapy3zx2y2}+\sqrt{10} \text{Mapyz2x2zx2y2}\right) $$ \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} $$ \frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $
$ {Y_{3}^{(3)}} $$ \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $$ \frac{i \left(\sqrt{15} \text{Mappxyzx3}-3 \text{Mappxyzxy2z2}-\sqrt{15} \text{Mapy3zx2y2}-3 \text{Mapyz2x2zx2y2}\right)}{4 \sqrt{6}} $$ \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}+2 \text{Mappx3xy2z2}-2 \text{Mapy3yz2x2}\right) $$ -\frac{1}{4} i \left(\sqrt{5} \text{Mapy3z3}+\sqrt{3} \text{Mapz3yz2x2}\right) $$ \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eapy3}-\sqrt{15} \text{Eapyz2x2}-2 (\text{Mappx3xy2z2}+\text{Mapy3yz2x2})\right) $$ -\frac{1}{8} i \left(\sqrt{10} \text{Mappxyzx3}-\sqrt{6} \text{Mappxyzxy2z2}+\sqrt{10} \text{Mapy3zx2y2}+\sqrt{6} \text{Mapyz2x2zx2y2}\right) $$ \frac{1}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eapy3}+3 \text{Eapyz2x2}+2 \sqrt{15} (\text{Mapy3yz2x2}-\text{Mappx3xy2z2})\right) $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ f_{\text{xyz}} $ $ f_{y\left(5y^2-r^2\right)} $ $ f_{z\left(5z^2-r^2\right)} $ $ f_{x\left(5x^2-r^2\right)} $ $ f_{y\left(z^2-x^2\right)} $ $ f_{z\left(x^2-y^2\right)} $ $ f_{x\left(y^2-z^2\right)} $
$ f_{\text{xyz}} $$ \text{Eappxyz} $$ 0 $$ 0 $$ \text{Mappxyzx3} $$ 0 $$ 0 $$ \text{Mappxyzxy2z2} $
$ f_{y\left(5y^2-r^2\right)} $$ 0 $$ \text{Eapy3} $$ \text{Mapy3z3} $$ 0 $$ \text{Mapy3yz2x2} $$ \text{Mapy3zx2y2} $$ 0 $
$ f_{z\left(5z^2-r^2\right)} $$ 0 $$ \text{Mapy3z3} $$ \text{Eapz3} $$ 0 $$ \text{Mapz3yz2x2} $$ \text{Mapz3zx2y2} $$ 0 $
$ f_{x\left(5x^2-r^2\right)} $$ \text{Mappxyzx3} $$ 0 $$ 0 $$ \text{Eappx3} $$ 0 $$ 0 $$ \text{Mappx3xy2z2} $
$ f_{y\left(z^2-x^2\right)} $$ 0 $$ \text{Mapy3yz2x2} $$ \text{Mapz3yz2x2} $$ 0 $$ \text{Eapyz2x2} $$ \text{Mapyz2x2zx2y2} $$ 0 $
$ f_{z\left(x^2-y^2\right)} $$ 0 $$ \text{Mapy3zx2y2} $$ \text{Mapz3zx2y2} $$ 0 $$ \text{Mapyz2x2zx2y2} $$ \text{Eapzx2y2} $$ 0 $
$ f_{x\left(y^2-z^2\right)} $$ \text{Mappxyzxy2z2} $$ 0 $$ 0 $$ \text{Mappx3xy2z2} $$ 0 $$ 0 $$ \text{Eappxy2z2} $

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

$$\text{Eappxyz}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$
$$\text{Eapy3}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$
$$\text{Eapz3}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$
$$\text{Eappx3}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$
$$\text{Eapyz2x2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$
$$\text{Eapzx2y2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$
$$\text{Eappxy2z2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-p orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Cs_X.Quanty
Akm = {{1, 0, A(1,0)} , 
       {1,-1, (I)*(B(1,1))} , 
       {1, 1, (I)*(B(1,1))} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ p_y $ $ p_z $ $ p_x $
$ \text{s} $$ \sqrt{\frac{2}{3}} B(1,1) $$ \frac{A(1,0)}{\sqrt{3}} $$ 0 $

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

Input format suitable for Quanty

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

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

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

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

Input format suitable for Quanty

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

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

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

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

Input format suitable for Quanty

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

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{-2}^{(2)}} $ $ {Y_{-1}^{(2)}} $ $ {Y_{0}^{(2)}} $ $ {Y_{1}^{(2)}} $ $ {Y_{2}^{(2)}} $
$ {Y_{-1}^{(1)}} $$ -\frac{1}{35} i \left(7 \sqrt{10} B(1,1)-\sqrt{15} B(3,1)\right) $$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $$ \frac{3}{7} i \sqrt{\frac{2}{5}} B(3,1)-\frac{i B(1,1)}{\sqrt{15}} $$ -\frac{1}{7} \sqrt{6} A(3,2) $$ \frac{3}{7} i B(3,3) $
$ {Y_{0}^{(1)}} $$ \frac{1}{7} \sqrt{3} A(3,2) $$ -\frac{i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} $$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $$ -\frac{i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} $$ \frac{1}{7} \sqrt{3} A(3,2) $
$ {Y_{1}^{(1)}} $$ \frac{3}{7} i B(3,3) $$ -\frac{1}{7} \sqrt{6} A(3,2) $$ \frac{3}{7} i \sqrt{\frac{2}{5}} B(3,1)-\frac{i B(1,1)}{\sqrt{15}} $$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $$ -\frac{1}{35} i \left(7 \sqrt{10} B(1,1)-\sqrt{15} B(3,1)\right) $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ d_{y^2-z^2} $ $ d_{3x^2-r^2} $ $ d_{\text{xz}} $ $ d_{\text{xy}} $ $ d_{\text{yz}} $
$ p_y $$ \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} B(3,1)-\frac{3}{14} B(3,3) $$ \frac{1}{210} \left(-14 \sqrt{30} B(1,1)-9 \sqrt{5} B(3,1)+45 \sqrt{3} B(3,3)\right) $$ 0 $$ 0 $$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) $
$ p_z $$ \frac{1}{70} \left(-\sqrt{5} (14 A(1,0)+9 A(3,0))-5 \sqrt{6} A(3,2)\right) $$ \frac{3 A(3,2)}{7 \sqrt{2}}-\frac{14 A(1,0)+9 A(3,0)}{14 \sqrt{15}} $$ 0 $$ 0 $$ \sqrt{\frac{2}{5}} B(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} B(3,1) $
$ p_x $$ 0 $$ 0 $$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right) $$ \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} B(3,1)+\frac{3}{7} B(3,3) $$ 0 $

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

Input format suitable for Quanty

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

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

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

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ f_{\text{xyz}} $ $ f_{y\left(5y^2-r^2\right)} $ $ f_{z\left(5z^2-r^2\right)} $ $ f_{x\left(5x^2-r^2\right)} $ $ f_{y\left(z^2-x^2\right)} $ $ f_{z\left(x^2-y^2\right)} $ $ f_{x\left(y^2-z^2\right)} $
$ p_y $$ 0 $$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $$ \frac{1}{45} \sqrt{\frac{2}{7}} \left(5 \sqrt{30} B(4,1)-27 B(2,1)\right) $$ 0 $$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $$ \frac{1}{21} \left(\sqrt{7} B(4,1)+7 B(4,3)\right)-\sqrt{\frac{6}{35}} B(2,1) $$ 0 $
$ p_z $$ 0 $$ \frac{1}{630} \left(-54 \sqrt{14} B(2,1)-5 \sqrt{15} \left(3 \sqrt{7} B(4,1)+7 B(4,3)\right)\right) $$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $$ 0 $$ \sqrt{\frac{6}{35}} B(2,1)+\frac{5 B(4,1)}{6 \sqrt{7}}-\frac{1}{6} B(4,3) $$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $$ 0 $
$ p_x $$ \sqrt{\frac{6}{35}} B(2,1)-\frac{B(4,1)}{3 \sqrt{7}}+\frac{1}{3} B(4,3) $$ 0 $$ 0 $$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $$ 0 $$ 0 $$ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

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

Input format suitable for Quanty

Input format suitable for Quanty

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

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{-3}^{(3)}} $ $ {Y_{-2}^{(3)}} $ $ {Y_{-1}^{(3)}} $ $ {Y_{0}^{(3)}} $ $ {Y_{1}^{(3)}} $ $ {Y_{2}^{(3)}} $ $ {Y_{3}^{(3)}} $
$ {Y_{-2}^{(2)}} $$ -\frac{1}{231} i \left(33 \sqrt{21} B(1,1)-11 \sqrt{14} B(3,1)+\sqrt{35} B(5,1)\right) $$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $$ -\frac{i \left(33 \sqrt{35} B(1,1)-22 \sqrt{210} B(3,1)+25 \sqrt{21} B(5,1)\right)}{1155} $$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $$ \frac{1}{231} i \sqrt{2} \left(11 \sqrt{7} B(3,3)-35 B(5,3)\right) $$ \frac{1}{11} \sqrt{10} A(5,4) $$ -\frac{5}{11} i \sqrt{\frac{2}{3}} B(5,5) $
$ {Y_{-1}^{(2)}} $$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $$ -\frac{1}{231} i \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)-2 \sqrt{210} B(5,1)\right) $$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $$ -\frac{i \left(33 \sqrt{15} B(1,1)-11 \sqrt{10} B(3,1)-100 B(5,1)\right)}{165 \sqrt{7}} $$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)+28 B(5,3)\right) $$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $
$ {Y_{0}^{(2)}} $$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)-14 B(5,3)\right) $$ \frac{1}{11} \sqrt{5} A(5,2) $$ -\frac{1}{385} i \left(11 \sqrt{210} B(1,1)+11 \sqrt{35} B(3,1)+25 \sqrt{14} B(5,1)\right) $$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $$ -\frac{1}{385} i \left(11 \sqrt{210} B(1,1)+11 \sqrt{35} B(3,1)+25 \sqrt{14} B(5,1)\right) $$ \frac{1}{11} \sqrt{5} A(5,2) $$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)-14 B(5,3)\right) $
$ {Y_{1}^{(2)}} $$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $$ \frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)+28 B(5,3)\right) $$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $$ -\frac{i \left(33 \sqrt{15} B(1,1)-11 \sqrt{10} B(3,1)-100 B(5,1)\right)}{165 \sqrt{7}} $$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $$ -\frac{1}{231} i \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)-2 \sqrt{210} B(5,1)\right) $$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $
$ {Y_{2}^{(2)}} $$ -\frac{5}{11} i \sqrt{\frac{2}{3}} B(5,5) $$ \frac{1}{11} \sqrt{10} A(5,4) $$ \frac{1}{231} i \sqrt{2} \left(11 \sqrt{7} B(3,3)-35 B(5,3)\right) $$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $$ -\frac{i \left(33 \sqrt{35} B(1,1)-22 \sqrt{210} B(3,1)+25 \sqrt{21} B(5,1)\right)}{1155} $$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $$ -\frac{1}{231} i \left(33 \sqrt{21} B(1,1)-11 \sqrt{14} B(3,1)+\sqrt{35} B(5,1)\right) $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ f_{\text{xyz}} $ $ f_{y\left(5y^2-r^2\right)} $ $ f_{z\left(5z^2-r^2\right)} $ $ f_{x\left(5x^2-r^2\right)} $ $ f_{y\left(z^2-x^2\right)} $ $ f_{z\left(x^2-y^2\right)} $ $ f_{x\left(y^2-z^2\right)} $
$ d_{y^2-z^2} $$ 0 $$ 3 \sqrt{\frac{3}{70}} B(1,1)-\frac{B(3,1)}{6 \sqrt{35}}-\frac{1}{2} \sqrt{\frac{3}{7}} B(3,3)+\frac{5 B(5,1)}{6 \sqrt{14}}+\frac{5}{44} \sqrt{3} B(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} B(5,5) $$ \frac{5 \sqrt{2} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)-\sqrt{105} (99 A(1,0)+44 A(3,0)+50 A(5,0))}{2310} $$ 0 $$ \frac{1}{924} \left(-66 \sqrt{14} B(1,1)-22 \sqrt{21} B(3,1)-22 \sqrt{35} B(3,3)-17 \sqrt{210} B(5,1)+7 \sqrt{5} B(5,3)+105 B(5,5)\right) $$ \frac{\sqrt{14} (-33 A(1,0)+22 A(3,0)-5 A(5,0))-42 \sqrt{15} A(5,2)-42 \sqrt{5} A(5,4)}{462 \sqrt{2}} $$ 0 $
$ d_{3x^2-r^2} $$ 0 $$ \frac{-198 \sqrt{70} B(1,1)+154 \sqrt{105} B(3,1)-5 \left(22 \sqrt{7} B(3,3)-5 \sqrt{42} B(5,1)+35 \left(B(5,3)+3 \sqrt{5} B(5,5)\right)\right)}{4620} $$ \frac{-\sqrt{35} (99 A(1,0)+44 A(3,0)+50 A(5,0))-5 \sqrt{6} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)}{2310} $$ 0 $$ \frac{1}{924} \left(-66 \sqrt{42} B(1,1)-66 \sqrt{7} B(3,1)-22 \sqrt{105} B(3,3)-9 \sqrt{70} B(5,1)+49 \sqrt{15} B(5,3)-105 \sqrt{3} B(5,5)\right) $$ \frac{\sqrt{42} (33 A(1,0)-22 A(3,0)+5 A(5,0))-42 \sqrt{5} A(5,2)+42 \sqrt{15} A(5,4)}{462 \sqrt{2}} $$ 0 $
$ d_{\text{xz}} $$ \frac{1}{231} \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)-2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $$ 0 $$ 0 $$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $$ 0 $$ 0 $$ \frac{1}{462} \left(-66 \sqrt{7} A(1,0)-11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)+25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)-21 \sqrt{10} A(5,4)\right) $
$ d_{\text{xy}} $$ \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right) $$ 0 $$ 0 $$ \frac{-66 \sqrt{210} B(1,1)-11 \sqrt{35} B(3,1)+5 \left(11 \sqrt{21} B(3,3)+5 \sqrt{14} B(5,1)-35 \sqrt{3} B(5,3)+35 \sqrt{15} B(5,5)\right)}{2310} $$ 0 $$ 0 $$ \frac{1}{462} \left(66 \sqrt{14} B(1,1)-33 \sqrt{21} B(3,1)+11 \sqrt{35} B(3,3)+3 \sqrt{210} B(5,1)-35 \sqrt{5} B(5,3)-105 B(5,5)\right) $
$ d_{\text{yz}} $$ 0 $$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $$ -\sqrt{\frac{6}{35}} B(1,1)+\frac{2 B(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} B(5,1) $$ 0 $$ \frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right) $$ \frac{1}{231} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)+2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $$ 0 $

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups C1 Cs Ci
Cn groups C2 C3 C4 C5 C6 C7 C8
Dn groups D2 D3 D4 D5 D6 D7 D8
Cnv groups C2v C3v C4v C5v C6v C7v C8v
Cnh groups C2h C3h C4h C5h C6h
Dnh groups D2h D3h D4h D5h D6h D7h D8h
Dnd groups D2d D3d D4d D5d D6d D7d D8d
Sn groups S2 S4 S6 S8 S10 S12
Cubic groups T Th Td O Oh I Ih
Linear groups C$\infty$v D$\infty$h

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