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physics_chemistry:point_groups:cs:orientation_z [2018/03/24 00:31] – Maurits W. Haverkortphysics_chemistry:point_groups:cs:orientation_z [2018/04/06 09:16] (current) – Maurits W. Haverkort
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 +~~CLOSETOC~~
 +
 ====== Orientation Z ====== ====== Orientation Z ======
  
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 ### ###
  
-  * [[physics_chemistry:point_groups:c1:orientation_|Point Group C1 with orientation ]]+  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
  
 ### ###
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   * [[physics_chemistry:point_groups:d3h:orientation_zy|Point Group D3h with orientation Zy]]   * [[physics_chemistry:point_groups:d3h:orientation_zy|Point Group D3h with orientation Zy]]
   * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]   * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]]
 +  * [[physics_chemistry:point_groups:d5h:orientation_zx|Point Group D5h with orientation Zx]]
 +  * [[physics_chemistry:point_groups:d5h:orientation_zy|Point Group D5h with orientation Zy]]
   * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]]   * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]]
   * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]]   * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]]
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 ### ###
  
-Any potential (function) can be written in spherical coordinates as a sum over spherical harmonics $$V(\vec{r}) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$. With C(m)k(θ,ϕ) a renormalised spherical harmonics C(m)k(θ,ϕ)=√4π2k+1Y(m)k(θ,ϕ). The allowed expansion coefficients $A_{k,m}(r)$, or once evaluated for a given radial wave-function $A_{k,m}=\langle\psi(r)|A_{k,m}(r)|\psi(r)\rangle$, such that $V(\vec{r}) is invariant under all symmetry operations of the Cs Point group with orientation Z are:+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 Ak,m(r) is a radial function and C(m)k(θ,ϕ) a renormalised spherical harmonics. $C(m)k(θ,ϕ)=√4π2k+1Y(m)k(θ,ϕ)$ 
 +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 Z the form of the expansion coefficients is:
  
 ### ###
  
-==== Input format suitable for Mathematica (Quanty.nb) ====+==== Expansion ====
  
 ### ###
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  $$A_{k,m} = \begin{cases}  $$A_{k,m} = \begin{cases}
  A(0,0) & k=0\land m=0 \\  A(0,0) & k=0\land m=0 \\
- -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ + -A(1,1)+i B(1,1) & k=1\land m=-1 \\ 
- A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ + A(1,1)+i B(1,1) & k=1\land m=1 \\ 
- A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\+ A(2,2)-i B(2,2) & k=2\land m=-2 \\
  A(2,0) & k=2\land m=0 \\  A(2,0) & k=2\land m=0 \\
- A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ + A(2,2)+i B(2,2) & k=2\land m=2 \\ 
- -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ + -A(3,3)+i B(3,3) & k=3\land m=-3 \\ 
- -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ + -A(3,1)+i B(3,1) & k=3\land m=-1 \\ 
- A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ + A(3,1)+i B(3,1) & k=3\land m=1 \\ 
- A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ + A(3,3)+i B(3,3) & k=3\land m=3 \\ 
- A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ + A(4,4)-i B(4,4) & k=4\land m=-4 \\ 
- A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\+ A(4,2)-i B(4,2) & k=4\land m=-2 \\
  A(4,0) & k=4\land m=0 \\  A(4,0) & k=4\land m=0 \\
- A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ + A(4,2)+i B(4,2) & k=4\land m=2 \\ 
- A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ + A(4,4)+i B(4,4) & k=4\land m=4 \\ 
- -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ + -A(5,5)+i B(5,5) & k=5\land m=-5 \\ 
- -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ + -A(5,3)+i B(5,3) & k=5\land m=-3 \\ 
- -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ + -A(5,1)+i B(5,1) & k=5\land m=-1 \\ 
- A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ + A(5,1)+i B(5,1) & k=5\land m=1 \\ 
- A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ + A(5,3)+i B(5,3) & k=5\land m=3 \\ 
- A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ + A(5,5)+i B(5,5) & k=5\land m=5 \\ 
- A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ + A(6,6)-i B(6,6) & k=6\land m=-6 \\ 
- A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ + A(6,4)-i B(6,4) & k=6\land m=-4 \\ 
- A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\+ A(6,2)-i B(6,2) & k=6\land m=-2 \\
  A(6,0) & k=6\land m=0 \\  A(6,0) & k=6\land m=0 \\
- A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ + A(6,2)+i B(6,2) & k=6\land m=2 \\ 
- A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ + A(6,4)+i B(6,4) & k=6\land m=4 \\ 
- A(6,6)+i \text{Ap}(6,6) & k=6\land m=6+ A(6,6)+i B(6,6) & k=6\land m=6
 \end{cases}$$ \end{cases}$$
 +
 +###
 +
 +==== Input format suitable for Mathematica (Quanty.nb) ====
 +
 +###
 +
 +<code Quanty Akm_Cs_Z.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]
 +
 +</code>
  
 ### ###
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 Akm = {{0, 0, A(0,0)} ,  Akm = {{0, 0, A(0,0)} , 
-       {1,-1, (-1)*(A(1,1)) + ((+1*I))*(Ap(1,1))} ,  +       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,  
-       {1, 1, A(1,1) + ((+1*I))*(Ap(1,1))} , +       {1, 1, A(1,1) + (I)*(B(1,1))} , 
        {2, 0, A(2,0)} ,         {2, 0, A(2,0)} , 
-       {2,-2, A(2,2) + ((+-1*I))*(Ap(2,2))} ,  +       {2,-2, A(2,2) + (-I)*(B(2,2))} ,  
-       {2, 2, A(2,2) + ((+1*I))*(Ap(2,2))} ,  +       {2, 2, A(2,2) + (I)*(B(2,2))} ,  
-       {3,-1, (-1)*(A(3,1)) + ((+1*I))*(Ap(3,1))} ,  +       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,  
-       {3, 1, A(3,1) + ((+1*I))*(Ap(3,1))} ,  +       {3, 1, A(3,1) + (I)*(B(3,1))} ,  
-       {3,-3, (-1)*(A(3,3)) + ((+1*I))*(Ap(3,3))} ,  +       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,  
-       {3, 3, A(3,3) + ((+1*I))*(Ap(3,3))} , +       {3, 3, A(3,3) + (I)*(B(3,3))} , 
        {4, 0, A(4,0)} ,         {4, 0, A(4,0)} , 
-       {4,-2, A(4,2) + ((+-1*I))*(Ap(4,2))} ,  +       {4,-2, A(4,2) + (-I)*(B(4,2))} ,  
-       {4, 2, A(4,2) + ((+1*I))*(Ap(4,2))} ,  +       {4, 2, A(4,2) + (I)*(B(4,2))} ,  
-       {4,-4, A(4,4) + ((+-1*I))*(Ap(4,4))} ,  +       {4,-4, A(4,4) + (-I)*(B(4,4))} ,  
-       {4, 4, A(4,4) + ((+1*I))*(Ap(4,4))} ,  +       {4, 4, A(4,4) + (I)*(B(4,4))} ,  
-       {5,-1, (-1)*(A(5,1)) + ((+1*I))*(Ap(5,1))} ,  +       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,  
-       {5, 1, A(5,1) + ((+1*I))*(Ap(5,1))} ,  +       {5, 1, A(5,1) + (I)*(B(5,1))} ,  
-       {5,-3, (-1)*(A(5,3)) + ((+1*I))*(Ap(5,3))} ,  +       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,  
-       {5, 3, A(5,3) + ((+1*I))*(Ap(5,3))} ,  +       {5, 3, A(5,3) + (I)*(B(5,3))} ,  
-       {5,-5, (-1)*(A(5,5)) + ((+1*I))*(Ap(5,5))} ,  +       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,  
-       {5, 5, A(5,5) + ((+1*I))*(Ap(5,5))} , +       {5, 5, A(5,5) + (I)*(B(5,5))} , 
        {6, 0, A(6,0)} ,         {6, 0, A(6,0)} , 
-       {6,-2, A(6,2) + ((+-1*I))*(Ap(6,2))} ,  +       {6,-2, A(6,2) + (-I)*(B(6,2))} ,  
-       {6, 2, A(6,2) + ((+1*I))*(Ap(6,2))} ,  +       {6, 2, A(6,2) + (I)*(B(6,2))} ,  
-       {6,-4, A(6,4) + ((+-1*I))*(Ap(6,4))} ,  +       {6,-4, A(6,4) + (-I)*(B(6,4))} ,  
-       {6, 4, A(6,4) + ((+1*I))*(Ap(6,4))} ,  +       {6, 4, A(6,4) + (I)*(B(6,4))} ,  
-       {6,-6, A(6,6) + ((+-1*I))*(Ap(6,6))} ,  +       {6,-6, A(6,6) + (-I)*(B(6,6))} ,  
-       {6, 6, A(6,6) + ((+1*I))*(Ap(6,6))} }+       {6, 6, A(6,6) + (I)*(B(6,6))} }
  
 </code> </code>
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 ### ###
  
-|  $  $  ^  $ \text{Subsuperscript}[\text{Y},0,\text{(0)}] $  ^  $ \text{Subsuperscript}[\text{Y},-1,\text{(1)}] $  ^  $ \text{Subsuperscript}[\text{Y},0,\text{(1)}] $  ^  $ \text{Subsuperscript}[\text{Y},1,\text{(1)}] $  ^  $ \text{Subsuperscript}[\text{Y},-2,\text{(2)}] ^ \text{Subsuperscript}[\text{Y},-1,\text{(2)}] ^ \text{Subsuperscript}[\text{Y},0,\text{(2)}] ^ \text{Subsuperscript}[\text{Y},1,\text{(2)}] ^ \text{Subsuperscript}[\text{Y},2,\text{(2)}] ^ \text{Subsuperscript}[\text{Y},-3,\text{(3)}] ^ \text{Subsuperscript}[\text{Y},-2,\text{(3)}] ^ \text{Subsuperscript}[\text{Y},-1,\text{(3)}] ^ \text{Subsuperscript}[\text{Y},0,\text{(3)}] ^ \text{Subsuperscript}[\text{Y},1,\text{(3)}] ^ \text{Subsuperscript}[\text{Y},2,\text{(3)}] ^ \text{Subsuperscript}[\text{Y},3,\text{(3)}] $  ^ +The operator representing the potential in second quantisation is given as: 
-^ $ \text{Subsuperscript}[\text{Y},0,\text{(0)}] $ |  Ass(0,0) |  −Asp(1,1)+iBsp(1,1)√3 |  0 |  −−Asp(1,1)+iBsp(1,1)√3 |  Asd(2,2)+iBsd(2,2)√5 |  0 |  Asd(2,0)√5 |  0 |  Asd(2,2)−iBsd(2,2)√5 |  −Asf(3,3)+iBsf(3,3)√7 |  0 |  −Asf(3,1)+iBsf(3,1)√7 |  0 |  −−Asf(3,1)+iBsf(3,1)√7 |  0 |  −−Asf(3,3)+iBsf(3,3)√7 | +$$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ 
-^ $ \text{Subsuperscript}[\text{Y},-1,\text{(1)}] $ |  −Asp(1,1)+iBsp(1,1)√3 |  App(0,0)−15App(2,0) |  0 |  −15√6(App(2,2)−iBpp(2,2)) |  17√35(Apd(3,1)+iBpd(3,1))−√25(Apd(1,1)+iBpd(1,1)) |  0 |  37√25(−Apd(3,1)+iBpd(3,1))−−Apd(1,1)+iBpd(1,1)√15 |  0 |  37(−Apd(3,3)+iBpd(3,3)) |  3(Apf(2,2)+iBpf(2,2))√35−Apf(4,2)+iBpf(4,2)3√21 |  0 |  35√27Apf(2,0)−13√27Apf(4,0) |  0 |  15√37(Apf(2,2)−iBpf(2,2))−13√57(Apf(4,2)−iBpf(4,2)) |  0 |  −2(Apf(4,4)−iBpf(4,4))3√3 | +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. 
-^ $ \text{Subsuperscript}[\text{Y},0,\text{(1)}] $ |  0 |  0 |  App(0,0)+25App(2,0) |  0 |  0 |  −Apd(1,1)+iBpd(1,1)√5−27√65(Apd(3,1)+iBpd(3,1)) |  0 |  −−Apd(1,1)+iBpd(1,1)√5−27√65(−Apd(3,1)+iBpd(3,1)) |  0 |  0 |  √335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7 |  0 |  35√37Apf(2,0)+4Apf(4,0)3√21 |  0 |  √335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√7 |  0 | +$$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ 
-^ $ \text{Subsuperscript}[\text{Y},1,\text{(1)}] $ |  Asp(1,1)+iBsp(1,1)√3 |  −15√6(App(2,2)+iBpp(2,2)) |  0 |  App(0,0)−15App(2,0) |  37(Apd(3,3)+iBpd(3,3)) |  0 |  37√25(Apd(3,1)+iBpd(3,1))−Apd(1,1)+iBpd(1,1)√15 |  0 |  17√35(−Apd(3,1)+iBpd(3,1))−√25(−Apd(1,1)+iBpd(1,1)) |  −2(Apf(4,4)+iBpf(4,4))3√3 |  0 |  15√37(Apf(2,2)+iBpf(2,2))−13√57(Apf(4,2)+iBpf(4,2)) |  0 |  35√27Apf(2,0)−13√27Apf(4,0) |  0 |  3(Apf(2,2)−iBpf(2,2))√35−Apf(4,2)−iBpf(4,2)3√21 | +Note the difference between the function $A_{k,m}andtheparameterA_{n''l'',n'l'}(k,m)$ 
-^ $ \text{Subsuperscript}[\text{Y},-2,\text{(2)}] $ |  Asd(2,2)−iBsd(2,2)√5 |  √25(−Apd(1,1)+iBpd(1,1))−17√35(−Apd(3,1)+iBpd(3,1)) |  0 |  −37(−Apd(3,3)+iBpd(3,3)) |  Add(0,0)−27Add(2,0)+121Add(4,0) |  0 |  17√53(Add(4,2)−iBdd(4,2))−27(Add(2,2)−iBdd(2,2)) |  0 |  13√107(Add(4,4)−iBdd(4,4)) |  −√37(Adf(1,1)+iBdf(1,1))+13√27(Adf(3,1)+iBdf(3,1))−133√57(Adf(5,1)+iBdf(5,1)) |  0 |  −−Adf(1,1)+iBdf(1,1)√35+2√2105(−Adf(3,1)+iBdf(3,1))−5(−Adf(5,1)+iBdf(5,1))11√21 |  0 |  13√27(−Adf(3,3)+iBdf(3,3))−533√2(−Adf(5,3)+iBdf(5,3)) |  0 |  −511√23(−Adf(5,5)+iBdf(5,5)) | + 
-^ $ \text{Subsuperscript}[\text{Y},-1,\text{(2)}] $ |  0 |  0 |  −Apd(1,1)+iBpd(1,1)√5+27√65(−Apd(3,1)+iBpd(3,1)) |  0 |  0 |  Add(0,0)+17Add(2,0)−421Add(4,0) |  0 |  −17√6(Add(2,2)−iBdd(2,2))−221√10(Add(4,2)−iBdd(4,2)) |  0 |  0 |  −√27(Adf(1,1)+iBdf(1,1))−Adf(3,1)+iBdf(3,1)√21+211√1021(Adf(5,1)+iBdf(5,1)) |  0 |  −√335(−Adf(1,1)+iBdf(1,1))+13√235(−Adf(3,1)+iBdf(3,1))+20(−Adf(5,1)+iBdf(5,1))33√7 |  0 |  13√57(−Adf(3,3)+iBdf(3,3))+433√5(−Adf(5,3)+iBdf(5,3)) |  0 | + 
-^ $ \text{Subsuperscript}[\text{Y},0,\text{(2)}] $ |  Asd(2,0)√5 |  Apd(1,1)+iBpd(1,1)√15−37√25(Apd(3,1)+iBpd(3,1)) |  0 |  −Apd(1,1)+iBpd(1,1)√15−37√25(−Apd(3,1)+iBpd(3,1)) |  17√53(Add(4,2)+iBdd(4,2))−27(Add(2,2)+iBdd(2,2)) |  0 |  Add(0,0)+27Add(2,0)+27Add(4,0) |  0 |  17√53(Add(4,2)−iBdd(4,2))−27(Add(2,2)−iBdd(2,2)) |  13√57(Adf(3,3)+iBdf(3,3))−233√5(Adf(5,3)+iBdf(5,3)) |  0 |  −√635(Adf(1,1)+iBdf(1,1))−Adf(3,1)+iBdf(3,1)√35−511√27(Adf(5,1)+iBdf(5,1)) |  0 |  −√635(−Adf(1,1)+iBdf(1,1))−−Adf(3,1)+iBdf(3,1)√35−511√27(−Adf(5,1)+iBdf(5,1)) |  0 |  13√57(−Adf(3,3)+iBdf(3,3))−233√5(−Adf(5,3)+iBdf(5,3)) | +### 
-^ $ \text{Subsuperscript}[\text{Y},1,\text{(2)}] $ |  0 |  0 |  Apd(1,1)+iBpd(1,1)√5+27√65(Apd(3,1)+iBpd(3,1)) |  0 |  0 |  −17√6(Add(2,2)+iBdd(2,2))−221√10(Add(4,2)+iBdd(4,2)) |  0 |  Add(0,0)+17Add(2,0)−421Add(4,0) |  0 |  0 |  13√57(Adf(3,3)+iBdf(3,3))+433√5(Adf(5,3)+iBdf(5,3)) |  0 |  −√335(Adf(1,1)+iBdf(1,1))+13√235(Adf(3,1)+iBdf(3,1))+20(Adf(5,1)+iBdf(5,1))33√7 |  0 |  −√27(−Adf(1,1)+iBdf(1,1))−−Adf(3,1)+iBdf(3,1)√21+211√1021(−Adf(5,1)+iBdf(5,1)) |  0 | + 
-^ $ \text{Subsuperscript}[\text{Y},2,\text{(2)}] $ |  Asd(2,2)+iBsd(2,2)√5 |  −37(Apd(3,3)+iBpd(3,3)) |  0 |  √25(Apd(1,1)+iBpd(1,1))−17√35(Apd(3,1)+iBpd(3,1)) |  13√107(Add(4,4)+iBdd(4,4)) |  0 |  17√53(Add(4,2)+iBdd(4,2))−27(Add(2,2)+iBdd(2,2)) |  0 |  Add(0,0)−27Add(2,0)+121Add(4,0) |  −511√23(Adf(5,5)+iBdf(5,5)) |  0 |  13√27(Adf(3,3)+iBdf(3,3))−533√2(Adf(5,3)+iBdf(5,3)) |  0 |  −Adf(1,1)+iBdf(1,1)√35+2√2105(Adf(3,1)+iBdf(3,1))−5(Adf(5,1)+iBdf(5,1))11√21 |  0 |  −√37(−Adf(1,1)+iBdf(1,1))+13√27(−Adf(3,1)+iBdf(3,1))−133√57(−Adf(5,1)+iBdf(5,1)) | + 
-^ $ \text{Subsuperscript}[\text{Y},-3,\text{(3)}] $ |  −Asf(3,3)+iBsf(3,3)√7 |  3(Apf(2,2)−iBpf(2,2))√35−Apf(4,2)−iBpf(4,2)3√21 |  0 |  −2(Apf(4,4)−iBpf(4,4))3√3 |  √37(−Adf(1,1)+iBdf(1,1))−13√27(−Adf(3,1)+iBdf(3,1))+133√57(−Adf(5,1)+iBdf(5,1)) |  0 |  233√5(−Adf(5,3)+iBdf(5,3))−13√57(−Adf(3,3)+iBdf(3,3)) |  0 |  511√23(−Adf(5,5)+iBdf(5,5)) |  Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0) |  0 |  −13√25(Aff(2,2)−iBff(2,2))+111√6(Aff(4,2)−iBff(4,2))−10429√7(Aff(6,2)−iBff(6,2)) |  0 |  111√143(Aff(4,4)−iBff(4,4))−5143√703(Aff(6,4)−iBff(6,4)) |  0 |  −1013√733(Aff(6,6)−iBff(6,6)) | + 
-^ $ \text{Subsuperscript}[\text{Y},-2,\text{(3)}] $ |  0 |  0 |  √335(Apf(2,2)−iBpf(2,2))+2(Apf(4,2)−iBpf(4,2))3√7 |  0 |  0 |  √27(−Adf(1,1)+iBdf(1,1))+−Adf(3,1)+iBdf(3,1)√21−211√1021(−Adf(5,1)+iBdf(5,1)) |  0 |  −13√57(−Adf(3,3)+iBdf(3,3))−433√5(−Adf(5,3)+iBdf(5,3)) |  0 |  0 |  Aff(0,0)−733Aff(4,0)+10143Aff(6,0) |  0 |  −2(Aff(2,2)−iBff(2,2))3√5−Aff(4,2)−iBff(4,2)11√3+20429√14(Aff(6,2)−iBff(6,2)) |  0 |  133√70(Aff(4,4)−iBff(4,4))+10143√14(Aff(6,4)−iBff(6,4)) |  0 | +### 
-^ $ \text{Subsuperscript}[\text{Y},-1,\text{(3)}] $ |  −Asf(3,1)+iBsf(3,1)√7 |  35√27Apf(2,0)−13√27Apf(4,0) |  0 |  15√37(Apf(2,2)−iBpf(2,2))−13√57(Apf(4,2)−iBpf(4,2)) |  Adf(1,1)+iBdf(1,1)√35−2√2105(Adf(3,1)+iBdf(3,1))+5(Adf(5,1)+iBdf(5,1))11√21 |  0 |  √635(−Adf(1,1)+iBdf(1,1))+−Adf(3,1)+iBdf(3,1)√35+511√27(−Adf(5,1)+iBdf(5,1)) |  0 |  533√2(−Adf(5,3)+iBdf(5,3))−13√27(−Adf(3,3)+iBdf(3,3)) |  −13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2)) |  0 |  Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0) |  0 |  −25√23(Aff(2,2)−iBff(2,2))−233√10(Aff(4,2)−iBff(4,2))−10143√353(Aff(6,2)−iBff(6,2)) |  0 |  111√143(Aff(4,4)−iBff(4,4))−5143√703(Aff(6,4)−iBff(6,4)) | + 
-^ $ \text{Subsuperscript}[\text{Y},0,\text{(3)}] $ |  0 |  0 |  35√37Apf(2,0)+4Apf(4,0)3√21 |  0 |  0 |  √335(Adf(1,1)+iBdf(1,1))−13√235(Adf(3,1)+iBdf(3,1))−20(Adf(5,1)+iBdf(5,1))33√7 |  0 |  √335(−Adf(1,1)+iBdf(1,1))−13√235(−Adf(3,1)+iBdf(3,1))−20(−Adf(5,1)+iBdf(5,1))33√7 |  0 |  0 |  −2(Aff(2,2)+iBff(2,2))3√5−Aff(4,2)+iBff(4,2)11√3+20429√14(Aff(6,2)+iBff(6,2)) |  0 |  Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0) |  0 |  −2(Aff(2,2)−iBff(2,2))3√5−Aff(4,2)−iBff(4,2)11√3+20429√14(Aff(6,2)−iBff(6,2)) |  0 | + 
-^ $ \text{Subsuperscript}[\text{Y},1,\text{(3)}] $ |  Asf(3,1)+iBsf(3,1)√7 |  15√37(Apf(2,2)+iBpf(2,2))−13√57(Apf(4,2)+iBpf(4,2)) |  0 |  35√27Apf(2,0)−13√27Apf(4,0) |  533√2(Adf(5,3)+iBdf(5,3))−13√27(Adf(3,3)+iBdf(3,3)) |  0 |  √635(Adf(1,1)+iBdf(1,1))+Adf(3,1)+iBdf(3,1)√35+511√27(Adf(5,1)+iBdf(5,1)) |  0 |  −Adf(1,1)+iBdf(1,1)√35−2√2105(−Adf(3,1)+iBdf(3,1))+5(−Adf(5,1)+iBdf(5,1))11√21 |  111√143(Aff(4,4)+iBff(4,4))−5143√703(Aff(6,4)+iBff(6,4)) |  0 |  −25√23(Aff(2,2)+iBff(2,2))−233√10(Aff(4,2)+iBff(4,2))−10143√353(Aff(6,2)+iBff(6,2)) |  0 |  Aff(0,0)+15Aff(2,0)+133Aff(4,0)−25143Aff(6,0) |  0 |  −13√25(Aff(2,2)−iBff(2,2))+111√6(Aff(4,2)−iBff(4,2))−10429√7(Aff(6,2)−iBff(6,2)) | +we can express the operator as  
-^ $ \text{Subsuperscript}[\text{Y},2,\text{(3)}] $ |  0 |  0 |  √335(Apf(2,2)+iBpf(2,2))+2(Apf(4,2)+iBpf(4,2))3√7 |  0 |  0 |  −13√57(Adf(3,3)+iBdf(3,3))−433√5(Adf(5,3)+iBdf(5,3)) |  0 |  √27(Adf(1,1)+iBdf(1,1))+Adf(3,1)+iBdf(3,1)√21−211√1021(Adf(5,1)+iBdf(5,1)) |  0 |  0 |  133√70(Aff(4,4)+iBff(4,4))+10143√14(Aff(6,4)+iBff(6,4)) |  0 |  −2(Aff(2,2)+iBff(2,2))3√5−Aff(4,2)+iBff(4,2)11√3+20429√14(Aff(6,2)+iBff(6,2)) |  0 |  Aff(0,0)−733Aff(4,0)+10143Aff(6,0) |  0 | +$$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ 
-^ $ \text{Subsuperscript}[\text{Y},3,\text{(3)}] $ |  Asf(3,3)+iBsf(3,3)√7 |  −2(Apf(4,4)+iBpf(4,4))3√3 |  0 |  3(Apf(2,2)+iBpf(2,2))√35−Apf(4,2)+iBpf(4,2)3√21 |  511√23(Adf(5,5)+iBdf(5,5)) |  0 |  233√5(Adf(5,3)+iBdf(5,3))−13√57(Adf(3,3)+iBdf(3,3)) |  0 |  √37(Adf(1,1)+iBdf(1,1))−13√27(Adf(3,1)+iBdf(3,1))+133√57(Adf(5,1)+iBdf(5,1)) |  −1013√733(Aff(6,6)+iBff(6,6)) |  0 |  111√143(Aff(4,4)+iBff(4,4))−5143√703(Aff(6,4)+iBff(6,4)) |  0 |  −13√25(Aff(2,2)+iBff(2,2))+111√6(Aff(4,2)+iBff(4,2))−10429√7(Aff(6,2)+iBff(6,2)) |  0 |  Aff(0,0)−13Aff(2,0)+111Aff(4,0)−5429Aff(6,0) |+ 
 + 
 +### 
 + 
 + 
 + 
 +### 
 + 
 + 
 +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 $A_{l'',l'}(k,m)canbecomplex.InsteadofallowingcomplexparameterswetookA_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (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{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }| \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} | 0 | \frac{\text{Asd}(2,0)}{\sqrt{5}} | 0 | \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} |\color{darkred}{ -\frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$| 
 +^$ {Y_{-1}^{(1)}} |\color{darkred}{ \frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }| \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) | 0 | -\frac{1}{5} \sqrt{6} (\text{App}(2,2)-i \text{Bpp}(2,2)) |\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}} }|\color{darkred}{ 0 }|\color{darkred}{ \frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }| \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} | 0 | \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) | 0 | -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $| 
 +^$ {Y_{0}^{(1)}} |\color{darkred}{ 0 }| 0 | \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) | 0 |\color{darkred}{ 0 }|\color{darkred}{ -\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }|\color{darkred}{ 0 }| 0 | \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} | 0 | \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} | 0 | \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} | 0 $| 
 +^$ {Y_{1}^{(1)}} |\color{darkred}{ \frac{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}} }| -\frac{1}{5} \sqrt{6} (\text{App}(2,2)+i \text{Bpp}(2,2)) | 0 | \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) |\color{darkred}{ \frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}} }|\color{darkred}{ 0 }|\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1)) }| -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) | 0 | \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) | 0 | \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $| 
 +^$ {Y_{-2}^{(2)}} | \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} |\color{darkred}{ \sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }| \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) | 0 | \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) | 0 | \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4)) |\color{darkred}{ -\sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }|\color{darkred}{ 0 }|\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$| 
 +^$ {Y_{-1}^{(2)}} | 0 |\color{darkred}{ 0 }|\color{darkred}{ \frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }|\color{darkred}{ 0 }| 0 | \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) | 0 | -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)-i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)-i \text{Bdd}(4,2)) | 0 |\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }|\color{darkred}{ 0 }|\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }$| 
 +^$ {Y_{0}^{(2)}} | \frac{\text{Asd}(2,0)}{\sqrt{5}} |\color{darkred}{ \frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1)) }| \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) | 0 | \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) | 0 | \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) |\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$| 
 +^$ {Y_{1}^{(2)}} | 0 |\color{darkred}{ 0 }|\color{darkred}{ \frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }|\color{darkred}{ 0 }| 0 | -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)+i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)+i \text{Bdd}(4,2)) | 0 | \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) | 0 |\color{darkred}{ 0 }|\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }|\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }$| 
 +^$ {Y_{2}^{(2)}} | \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} |\color{darkred}{ -\frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }|\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1)) }| \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4)) | 0 | \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) | 0 | \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) |\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }|\color{darkred}{ 0 }|\color{darkred}{ -\sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }$| 
 +^$ {Y_{-3}^{(3)}} |\color{darkred}{ \frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }| \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} | 0 | -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} |\color{darkred}{ \sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \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) | 0 | -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) | 0 | \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) | 0 | -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $| 
 +^$ {Y_{-2}^{(3)}} |\color{darkred}{ 0 }| 0 | \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} | 0 |\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }| 0 | \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) | 0 | -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) | 0 | \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4)) | 0 $| 
 +^$ {Y_{-1}^{(3)}} |\color{darkred}{ \frac{-\text{Asf}(3,1)+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) | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) |\color{darkred}{ \frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }|\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }| -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) | 0 | \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) | 0 | -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)-i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)-i \text{Bff}(6,2)) | 0 | \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $| 
 +^$ {Y_{0}^{(3)}} |\color{darkred}{ 0 }| 0 | \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} | 0 |\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }|\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}} }|\color{darkred}{ 0 }| 0 | -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) | 0 | \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) | 0 | -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) | 0 $| 
 +^$ {Y_{1}^{(3)}} |\color{darkred}{ \frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}} }| \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) | 0 | \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) |\color{darkred}{ \frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }|\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}} }| \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) | 0 | -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)+i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)+i \text{Bff}(6,2)) | 0 | \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) | 0 | -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $| 
 +^$ {Y_{2}^{(3)}} |\color{darkred}{ 0 }| 0 | \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} | 0 |\color{darkred}{ 0 }|\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }|\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }|\color{darkred}{ 0 }| 0 | \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4)) | 0 | -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) | 0 | \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) | 0 $| 
 +^$ {Y_{3}^{(3)}} |\color{darkred}{ \frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }| -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} | 0 | \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} |\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }|\color{darkred}{ 0 }|\color{darkred}{ \frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }|\color{darkred}{ 0 }|\color{darkred}{ \sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1)) }| -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) | 0 | \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) | 0 | -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) | 0 | \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|
  
  
Line 192: Line 244:
  
 ### ###
 +
 +
 +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(1)0  ^  Y(1)1  ^  Y(2)−2  ^  Y(2)−1  ^  Y(2)0  ^  Y(2)1  ^  Y(2)2  ^  Y(3)−3  ^  Y(3)−2  ^  Y(3)−1  ^  Y(3)0  ^  Y(3)1  ^  Y(3)2  ^  Y(3)3  ^
 +^s|1|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
 +^px|0|1√2|0|−1√2|0|0|0|0|0|0|0|0|0|0|0|0|
 +^py|0|i√2|0|i√2|0|0|0|0|0|0|0|0|0|0|0|0|
 +^pz|0|0|1|0|0|0|0|0|0|0|0|0|0|0|0|0|
 +^dx2−y2|0|0|0|0|1√2|0|0|0|1√2|0|0|0|0|0|0|0|
 +^d3z2−r2|0|0|0|0|0|0|1|0|0|0|0|0|0|0|0|0|
 +^dyz|0|0|0|0|0|i√2|0|i√2|0|0|0|0|0|0|0|0|
 +^dxz|0|0|0|0|0|1√2|0|−1√2|0|0|0|0|0|0|0|0|
 +^dxy|0|0|0|0|i√2|0|0|0|−i√2|0|0|0|0|0|0|0|
 +^fxyz|0|0|0|0|0|0|0|0|0|0|i√2|0|0|0|−i√2|0|
 +^fx(5x2−r2)|0|0|0|0|0|0|0|0|0|√54|0|−√34|0|√34|0|−√54|
 +^fy(5y2−r2)|0|0|0|0|0|0|0|0|0|−i√54|0|−i√34|0|−i√34|0|−i√54|
 +^fz(5z2−r2)|0|0|0|0|0|0|0|0|0|0|0|0|1|0|0|0|
 +^fx(y2−z2)|0|0|0|0|0|0|0|0|0|−√34|0|−√54|0|√54|0|√34|
 +^fy(z2−x2)|0|0|0|0|0|0|0|0|0|−i√34|0|i√54|0|i√54|0|−i√34|
 +^fz(x2−y2)|0|0|0|0|0|0|0|0|0|0|1√2|0|0|0|1√2|0|
  
  
Line 200: Line 279:
 ### ###
  
 +After rotation we find
  
 ### ###
  
-===== Potential for s orbitals ===== 
  
-===== Potential for p orbitals ===== 
  
-===== Potential for d orbitals =====+###
  
-===== Potential for f orbitals =====+|    ^  s  ^  px  ^  py  ^  pz  ^  dx2−y2  ^  d3z2−r2  ^  dyz  ^  dxz  ^  dxy  ^  fxyz  ^  fx(5x2−r2)  ^  fy(5y2−r2)  ^  fz(5z2−r2)  ^  fx(y2−z2)  ^  fy(z2−x2)  ^  fz(x2−y2)  ^ 
 +^s|Ass(0,0)|−√23Asp(1,1)|√23Bsp(1,1)|0|√25Asd(2,2)|Asd(2,0)√5|0|0|−√25Bsd(2,2)|0|12√37Asf(3,1)−12√57Asf(3,3)|−12√37Bsf(3,1)−12√57Bsf(3,3)|0|12√57Asf(3,1)+12√37Asf(3,3)|12√57Bsf(3,1)−12√37Bsf(3,3)|0| 
 +^px|−√23Asp(1,1)|App(0,0)−15App(2,0)+15√6App(2,2)|−15√6Bpp(2,2)|0|−√25Apd(1,1)+17√35Apd(3,1)−37Apd(3,3)|√215Apd(1,1)−6Apd(3,1)7√5|0|0|√25Bpd(1,1)−17√35Bpd(3,1)+37Bpd(3,3)|0|−310√37Apf(2,0)+9Apf(2,2)5√14+Apf(4,0)2√21−13√1021Apf(4,2)+13√56Apf(4,4)|35√27Bpf(2,2)+13√542Bpf(4,2)+13√56Bpf(4,4)|0|−3Apf(2,0)2√35−√370Apf(2,2)+16√57Apf(4,0)−13√27Apf(4,2)−Apf(4,4)3√2|√635Bpf(2,2)−Bpf(4,2)√14+Bpf(4,4)3√2|0| 
 +^py|√23Bsp(1,1)|−15√6Bpp(2,2)|App(0,0)−15App(2,0)−15√6App(2,2)|0|−√25Bpd(1,1)+17√35Bpd(3,1)+37Bpd(3,3)|6Bpd(3,1)7√5−√215Bpd(1,1)|0|0|−√25Apd(1,1)+17√35Apd(3,1)+37Apd(3,3)|0|35√27Bpf(2,2)+13√542Bpf(4,2)−13√56Bpf(4,4)|−310√37Apf(2,0)−9Apf(2,2)5√14+Apf(4,0)2√21+13√1021Apf(4,2)+13√56Apf(4,4)|0|−√635Bpf(2,2)+Bpf(4,2)√14+Bpf(4,4)3√2|3Apf(2,0)2√35−√370Apf(2,2)−16√57Apf(4,0)−13√27Apf(4,2)+Apf(4,4)3√2|0| 
 +^pz|0|0|0|App(0,0)+25App(2,0)|0|0|√25Bpd(1,1)+47√35Bpd(3,1)|−√25Apd(1,1)−47√35Apd(3,1)|0|−√635Bpf(2,2)−23√27Bpf(4,2)|0|0|35√37Apf(2,0)+4Apf(4,0)3√21|0|0|√635Apf(2,2)+23√27Apf(4,2)| 
 +^dx2−y2|√25Asd(2,2)|−√25Apd(1,1)+17√35Apd(3,1)−37Apd(3,3)|−√25Bpd(1,1)+17√35Bpd(3,1)+37Bpd(3,3)|0|Add(0,0)−27Add(2,0)+121Add(4,0)+13√107Add(4,4)|17√103Add(4,2)−27√2Add(2,2)|0|0|−13√107Bdd(4,4)|0|−3√370Adf(1,1)+11Adf(3,1)6√35−Adf(3,3)2√21−533√27Adf(5,1)+5Adf(5,3)22√3−522√53Adf(5,5)|−3√370Bdf(1,1)+11Bdf(3,1)6√35+Bdf(3,3)2√21−533√27Bdf(5,1)−5Bdf(5,3)22√3−522√53Bdf(5,5)|0|Adf(1,1)√14+Adf(3,1)2√21−16√57Adf(3,3)−111√1021Adf(5,1)+566√5Adf(5,3)+522Adf(5,5)|−Bdf(1,1)√14−Bdf(3,1)2√21−16√57Bdf(3,3)+111√1021Bdf(5,1)+566√5Bdf(5,3)−522Bdf(5,5)|0| 
 +^d3z2−r2|Asd(2,0)√5|√215Apd(1,1)−6Apd(3,1)7√5|6Bpd(3,1)7√5−√215Bpd(1,1)|0|17√103Add(4,2)−27√2Add(2,2)|Add(0,0)+27Add(2,0)+27Add(4,0)|0|0|27√2Bdd(2,2)−17√103Bdd(4,2)|0|3Adf(1,1)√70+12√335Adf(3,1)+5Adf(3,3)6√7+511√314Adf(5,1)−533Adf(5,3)|−3Bdf(1,1)√70−12√335Bdf(3,1)+5Bdf(3,3)6√7−511√314Bdf(5,1)−533Bdf(5,3)|0|√314Adf(1,1)+Adf(3,1)2√7−12√521Adf(3,3)+511√514Adf(5,1)+111√53Adf(5,3)|√314Bdf(1,1)+Bdf(3,1)2√7+12√521Bdf(3,3)+511√514Bdf(5,1)−111√53Bdf(5,3)|0| 
 +^dyz|0|0|0|√25Bpd(1,1)+47√35Bpd(3,1)|0|0|Add(0,0)+17Add(2,0)−17√6Add(2,2)−421Add(4,0)−221√10Add(4,2)|−17√6Bdd(2,2)−221√10Bdd(4,2)|0|−√27Adf(1,1)−Adf(3,1)√21+13√57Adf(3,3)+211√1021Adf(5,1)+433√5Adf(5,3)|0|0|−√635Bdf(1,1)+2Bdf(3,1)3√35+2033√27Bdf(5,1)|0|0|−√27Bdf(1,1)−Bdf(3,1)√21+13√57Bdf(3,3)+211√1021Bdf(5,1)+433√5Bdf(5,3)| 
 +^dxz|0|0|0|−√25Apd(1,1)−47√35Apd(3,1)|0|0|−17√6Bdd(2,2)−221√10Bdd(4,2)|Add(0,0)+17Add(2,0)+17√6Add(2,2)−421Add(4,0)+221√10Add(4,2)|0|√27Bdf(1,1)+Bdf(3,1)√21+13√57Bdf(3,3)−211√1021Bdf(5,1)+433√5Bdf(5,3)|0|0|√635Adf(1,1)−2Adf(3,1)3√35−2033√27Adf(5,1)|0|0|−√27Adf(1,1)−Adf(3,1)√21−13√57Adf(3,3)+211√1021Adf(5,1)−433√5Adf(5,3)| 
 +^dxy|−√25Bsd(2,2)|√25Bpd(1,1)−17√35Bpd(3,1)+37Bpd(3,3)|−√25Apd(1,1)+17√35Apd(3,1)+37Apd(3,3)|0|−13√107Bdd(4,4)|27√2Bdd(2,2)−17√103Bdd(4,2)|0|0|Add(0,0)−27Add(2,0)+121Add(4,0)−13√107Add(4,4)|0|−√635Bdf(1,1)−Bdf(3,1)6√35+Bdf(3,3)2√21+5Bdf(5,1)33√14−5Bdf(5,3)22√3+522√53Bdf(5,5)|√635Adf(1,1)+Adf(3,1)6√35+Adf(3,3)2√21−5Adf(5,1)33√14−5Adf(5,3)22√3−522√53Adf(5,5)|0|√27Bdf(1,1)−12√37Bdf(3,1)+16√57Bdf(3,3)+111√1514Bdf(5,1)−566√5Bdf(5,3)−522Bdf(5,5)|√27Adf(1,1)−12√37Adf(3,1)−16√57Adf(3,3)+111√1514Adf(5,1)+566√5Adf(5,3)−522Adf(5,5)|0| 
 +^fxyz|0|0|0|−√635Bpf(2,2)−23√27Bpf(4,2)|0|0|−√27Adf(1,1)−Adf(3,1)√21+13√57Adf(3,3)+211√1021Adf(5,1)+433√5Adf(5,3)|√27Bdf(1,1)+Bdf(3,1)√21+13√57Bdf(3,3)−211√1021Bdf(5,1)+433√5Bdf(5,3)|0|Aff(0,0)−733Aff(4,0)−133√70Aff(4,4)+10143Aff(6,0)−10143√14Aff(6,4)|0|0|23√25Bff(2,2)+111√23Bff(4,2)−40429√7Bff(6,2)|0|0|−133√70Bff(4,4)−10143√14Bff(6,4)| 
 +^fx(5x2−r2)|12√37Asf(3,1)−12√57Asf(3,3)|−310√37Apf(2,0)+9Apf(2,2)5√14+Apf(4,0)2√21−13√1021Apf(4,2)+13√56Apf(4,4)|35√27Bpf(2,2)+13√542Bpf(4,2)−13√56Bpf(4,4)|0|−3√370Adf(1,1)+11Adf(3,1)6√35−Adf(3,3)2√21−533√27Adf(5,1)+5Adf(5,3)22√3−522√53Adf(5,5)|3Adf(1,1)√70+12√335Adf(3,1)+5Adf(3,3)6√7+511√314Adf(5,1)−533Adf(5,3)|0|0|−√635Bdf(1,1)−Bdf(3,1)6√35+Bdf(3,3)2√21+5Bdf(5,1)33√14−5Bdf(5,3)22√3+522√53Bdf(5,5)|0|Aff(0,0)−215Aff(2,0)+25√23Aff(2,2)+344Aff(4,0)−111√52Aff(4,2)+122√352Aff(4,4)−125Aff(6,0)1716+25572√353Aff(6,2)−25286√72Aff(6,4)+2552√733Aff(6,6)|Bff(2,2)5√6−111√10Bff(4,2)−5572√353Bff(6,2)+2552√733Bff(6,6)|0|Aff(2,0)√15+13√25Aff(2,2)−144√53Aff(4,0)+Aff(4,2)11√6+122√76Aff(4,4)−35572√53Aff(6,0)+85√7Aff(6,2)1716−5286√356Aff(6,4)−552√3511Aff(6,6)|Bff(2,2)3√10+211√23Bff(4,2)+111√143Bff(4,4)+5132√7Bff(6,2)−5143√703Bff(6,4)+552√3511Bff(6,6)|0| 
 +^fy(5y2−r2)|−12√37Bsf(3,1)−12√57Bsf(3,3)|35√27Bpf(2,2)+13√542Bpf(4,2)+13√56Bpf(4,4)|−310√37Apf(2,0)−9Apf(2,2)5√14+Apf(4,0)2√21+13√1021Apf(4,2)+13√56Apf(4,4)|0|−3√370Bdf(1,1)+11Bdf(3,1)6√35+Bdf(3,3)2√21−533√27Bdf(5,1)−5Bdf(5,3)22√3−522√53Bdf(5,5)|−3Bdf(1,1)√70−12√335Bdf(3,1)+5Bdf(3,3)6√7−511√314Bdf(5,1)−533Bdf(5,3)|0|0|√635Adf(1,1)+Adf(3,1)6√35+Adf(3,3)2√21−5Adf(5,1)33√14−5Adf(5,3)22√3−522√53Adf(5,5)|0|Bff(2,2)5√6−111√10Bff(4,2)−5572√353Bff(6,2)+2552√733Bff(6,6)|Aff(0,0)−215Aff(2,0)−25√23Aff(2,2)+344Aff(4,0)+111√52Aff(4,2)+122√352Aff(4,4)−125Aff(6,0)1716−25572√353Aff(6,2)−25286√72Aff(6,4)−2552√733Aff(6,6)|0|−Bff(2,2)3√10−211√23Bff(4,2)+111√143Bff(4,4)−5132√7Bff(6,2)−5143√703Bff(6,4)−552√3511Bff(6,6)|−Aff(2,0)√15+13√25Aff(2,2)+144√53Aff(4,0)+Aff(4,2)11√6−122√76Aff(4,4)+35572√53Aff(6,0)+85√7Aff(6,2)1716+5286√356Aff(6,4)−552√3511Aff(6,6)|0| 
 +^fz(5z2−r2)|0|0|0|35√37Apf(2,0)+4Apf(4,0)3√21|0|0|−√635Bdf(1,1)+2Bdf(3,1)3√35+2033√27Bdf(5,1)|√635Adf(1,1)−2Adf(3,1)3√35−2033√27Adf(5,1)|0|23√25Bff(2,2)+111√23Bff(4,2)−40429√7Bff(6,2)|0|0|Aff(0,0)+415Aff(2,0)+211Aff(4,0)+100429Aff(6,0)|0|0|−23√25Aff(2,2)−111√23Aff(4,2)+40429√7Aff(6,2)| 
 +^fx(y2−z2)|12√57Asf(3,1)+12√37Asf(3,3)|−3Apf(2,0)2√35−√370Apf(2,2)+16√57Apf(4,0)−13√27Apf(4,2)−Apf(4,4)3√2|−√635Bpf(2,2)+Bpf(4,2)√14+Bpf(4,4)3√2|0|Adf(1,1)√14+Adf(3,1)2√21−16√57Adf(3,3)−111√1021Adf(5,1)+566√5Adf(5,3)+522Adf(5,5)|√314Adf(1,1)+Adf(3,1)2√7−12√521Adf(3,3)+511√514Adf(5,1)+111√53Adf(5,3)|0|0|√27Bdf(1,1)−12√37Bdf(3,1)+16√57Bdf(3,3)+111√1514Bdf(5,1)−566√5Bdf(5,3)−522Bdf(5,5)|0|Aff(2,0)√15+13√25Aff(2,2)−144√53Aff(4,0)+Aff(4,2)11√6+122√76Aff(4,4)−35572√53Aff(6,0)+85√7Aff(6,2)1716−5286√356Aff(6,4)−552√3511Aff(6,6)|−Bff(2,2)3√10−211√23Bff(4,2)+111√143Bff(4,4)−5132√7Bff(6,2)−5143√703Bff(6,4)−552√3511Bff(6,6)|0|Aff(0,0)+7132Aff(4,0)+733√52Aff(4,2)−122√352Aff(4,4)−544Aff(6,0)+5572√105Aff(6,2)+25286√72Aff(6,4)+552√2111Aff(6,6)|Bff(2,2)√6−133√10Bff(4,2)+35572√353Bff(6,2)−552√2111Bff(6,6)|0| 
 +^fy(z2−x2)|12√57Bsf(3,1)−12√37Bsf(3,3)|√635Bpf(2,2)−Bpf(4,2)√14+Bpf(4,4)3√2|3Apf(2,0)2√35−√370Apf(2,2)−16√57Apf(4,0)−13√27Apf(4,2)+Apf(4,4)3√2|0|−Bdf(1,1)√14−Bdf(3,1)2√21−16√57Bdf(3,3)+111√1021Bdf(5,1)+566√5Bdf(5,3)−522Bdf(5,5)|√314Bdf(1,1)+Bdf(3,1)2√7+12√521Bdf(3,3)+511√514Bdf(5,1)−111√53Bdf(5,3)|0|0|√27Adf(1,1)−12√37Adf(3,1)−16√57Adf(3,3)+111√1514Adf(5,1)+566√5Adf(5,3)−522Adf(5,5)|0|Bff(2,2)3√10+211√23Bff(4,2)+111√143Bff(4,4)+5132√7Bff(6,2)−5143√703Bff(6,4)+552√3511Bff(6,6)|−Aff(2,0)√15+13√25Aff(2,2)+144√53Aff(4,0)+Aff(4,2)11√6−122√76Aff(4,4)+35572√53Aff(6,0)+85√7Aff(6,2)1716+5286√356Aff(6,4)−552√3511Aff(6,6)|0|Bff(2,2)√6−133√10Bff(4,2)+35572√353Bff(6,2)−552√2111Bff(6,6)|Aff(0,0)+7132Aff(4,0)−733√52Aff(4,2)−122√352Aff(4,4)−544Aff(6,0)−5572√105Aff(6,2)+25286√72Aff(6,4)−552√2111Aff(6,6)|0| 
 +^fz(x2−y2)|0|0|0|√635Apf(2,2)+23√27Apf(4,2)|0|0|−√27Bdf(1,1)−Bdf(3,1)√21+13√57Bdf(3,3)+211√1021Bdf(5,1)+433√5Bdf(5,3)|−√27Adf(1,1)−Adf(3,1)√21−13√57Adf(3,3)+211√1021Adf(5,1)−433√5Adf(5,3)|0|−133√70Bff(4,4)−10143√14Bff(6,4)|0|0|−23√25Aff(2,2)−111√23Aff(4,2)+40429√7Aff(6,2)|0|0|Aff(0,0)−733Aff(4,0)+133√70Aff(4,4)+10143Aff(6,0)+10143√14Aff(6,4)|
  
-===== Potential for s-p orbital mixing ===== 
  
-===== Potential for s-d orbital mixing =====+###
  
-===== Potential for s-f orbital mixing =====+===== Coupling for a single shell =====
  
-===== Potential for p-d orbital mixing ===== 
  
-===== Potential for p-f orbital mixing ===== 
  
-===== Potential for d-f orbital mixing =====+### 
 + 
 +Although the parameters Al″ 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 <hiddenSwitch expand all>  
 + 
 +### 
 + 
 +==== Potential for s orbitals ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + \text{Ap} & k=0\land m=0 \\ 
 + 0 & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{Ap, k == 0 && m == 0}}, 0] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{0, 0, Ap} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** > 
 + 
 +### 
 + 
 +|      ^  {Y_{0}^{(0)}}   ^ 
 +^ {Y_{0}^{(0)}} | \text{Ap} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  \text{s}   ^ 
 +^ \text{s} | \text{Ap} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Rotation matrix used** > 
 + 
 +### 
 + 
 +|      ^  {Y_{0}^{(0)}}   ^ 
 +^ \text{s} | 1 | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Irriducible representations and their onsite energy** > 
 + 
 +### 
 + 
 +^ ^\text{Ap} | {{:physics_chemistry:pointgroup:cs_z_orb_0_1.png?150}} | 
 +|\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 }} | ::: | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for p orbitals ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + \frac{1}{3} (\text{Eapp}+\text{Eapx}+\text{Eapy}) & k=0\land m=0 \\ 
 + 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ 
 + \frac{5 (\text{Eapx}-\text{Eapy}+2 i \text{Mapxy})}{2 \sqrt{6}} & k=2\land m=-2 \\ 
 + \frac{5}{6} (2 \text{Eapp}-\text{Eapx}-\text{Eapy}) & k=2\land m=0 \\ 
 + \frac{5 (\text{Eapx}-\text{Eapy}-2 i \text{Mapxy})}{2 \sqrt{6}} & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{(Eapp + Eapx + Eapy)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != 0 && m != 2)}, {(5*(Eapx - Eapy + (2*I)*Mapxy))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(2*Eapp - Eapx - Eapy))/6, k == 2 && m == 0}}, (5*(Eapx - Eapy - (2*I)*Mapxy))/(2*Sqrt[6])] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{0, 0, (1/3)*(Eapp + Eapx + Eapy)} ,  
 +       {2, 0, (5/6)*((2)*(Eapp) + (-1)*(Eapx) + (-1)*(Eapy))} ,  
 +       {2, 2, (5/2)*((1/(sqrt(6)))*(Eapx + (-1)*(Eapy) + (-2*I)*(Mapxy)))} ,  
 +       {2,-2, (5/2)*((1/(sqrt(6)))*(Eapx + (-1)*(Eapy) + (2*I)*(Mapxy)))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** > 
 + 
 +### 
 + 
 +|      ^  {Y_{-1}^{(1)}}   ^  {Y_{0}^{(1)}}   ^  {Y_{1}^{(1)}}   ^ 
 +^ {Y_{-1}^{(1)}} | \frac{\text{Eapx}+\text{Eapy}}{2} | 0 | \frac{1}{2} (-\text{Eapx}+\text{Eapy}-2 i \text{Mapxy}) | 
 +^ {Y_{0}^{(1)}} | 0 | \text{Eapp} | 0 | 
 +^ {Y_{1}^{(1)}} | \frac{1}{2} (-\text{Eapx}+\text{Eapy}+2 i \text{Mapxy}) | 0 | \frac{\text{Eapx}+\text{Eapy}}{2} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  p_x   ^  p_y   ^  p_z   ^ 
 +^ p_x | \text{Eapx} | \text{Mapxy} | 0 | 
 +^ p_y | \text{Mapxy} | \text{Eapy} | 0 | 
 +^ p_z | 0 | 0 | \text{Eapp} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Rotation matrix used** > 
 + 
 +### 
 + 
 +|      ^  {Y_{-1}^{(1)}}   ^  {Y_{0}^{(1)}}   ^  {Y_{1}^{(1)}}   ^ 
 +^ p_x | \frac{1}{\sqrt{2}} | 0 | -\frac{1}{\sqrt{2}} | 
 +^ p_y | \frac{i}{\sqrt{2}} | 0 | \frac{i}{\sqrt{2}} | 
 +^ p_z | 0 | 1 | 0 | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Irriducible representations and their onsite energy** > 
 + 
 +### 
 + 
 +^ ^\text{Eapx} | {{:physics_chemistry:pointgroup:cs_z_orb_1_1.png?150}} | 
 +|\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi ) | ::: | 
 +|\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{2} \sqrt{\frac{3}{\pi }} x | ::: | 
 +^ ^\text{Eapy} | {{:physics_chemistry:pointgroup:cs_z_orb_1_2.png?150}} | 
 +|\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{Eapp} | {{:physics_chemistry:pointgroup:cs_z_orb_1_3.png?150}} | 
 +|\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 | ::: | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for d orbitals ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + \frac{1}{5} (\text{Eappxz}+\text{Eappyz}+\text{Eapx2y2}+\text{Eapxy}+\text{Eapz2}) & k=0\land m=0 \\ 
 + 0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ 
 + \frac{\sqrt{3} \text{Eappxz}-\sqrt{3} \text{Eappyz}+2 i \sqrt{3} \text{Mappyzxz}-4 \text{Mapx2y2z2}-4 i \text{Mapz2xy}}{2 \sqrt{2}} & k=2\land m=-2 \\ 
 + \frac{1}{2} (\text{Eappxz}+\text{Eappyz}-2 (\text{Eapx2y2}+\text{Eapxy}-\text{Eapz2})) & k=2\land m=0 \\ 
 + \frac{\sqrt{3} \text{Eappxz}-\sqrt{3} \text{Eappyz}-2 i \sqrt{3} \text{Mappyzxz}-4 \text{Mapx2y2z2}+4 i \text{Mapz2xy}}{2 \sqrt{2}} & k=2\land m=2 \\ 
 + \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eapx2y2}-\text{Eapxy}+2 i \text{Mapx2y2xy}) & k=4\land m=-4 \\ 
 + \frac{3 \left(\text{Eappxz}-\text{Eappyz}+2 i \text{Mappyzxz}+\sqrt{3} \text{Mapx2y2z2}+i \sqrt{3} \text{Mapz2xy}\right)}{\sqrt{10}} & k=4\land m=-2 \\ 
 + -\frac{3}{10} (4 \text{Eappxz}+4 \text{Eappyz}-\text{Eapx2y2}-\text{Eapxy}-6 \text{Eapz2}) & k=4\land m=0 \\ 
 + \frac{3 \left(\text{Eappxz}-\text{Eappyz}-2 i \text{Mappyzxz}+\sqrt{3} \text{Mapx2y2z2}-i \sqrt{3} \text{Mapz2xy}\right)}{\sqrt{10}} & k=4\land m=2 \\ 
 + \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eapx2y2}-\text{Eapxy}-2 i \text{Mapx2y2xy}) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{(Eappxz + Eappyz + Eapx2y2 + Eapxy + Eapz2)/5, k == 0 && m == 0}, {0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {(Sqrt[3]*Eappxz - Sqrt[3]*Eappyz + (2*I)*Sqrt[3]*Mappyzxz - 4*Mapx2y2z2 - (4*I)*Mapz2xy)/(2*Sqrt[2]), k == 2 && m == -2}, {(Eappxz + Eappyz - 2*(Eapx2y2 + Eapxy - Eapz2))/2, k == 2 && m == 0}, {(Sqrt[3]*Eappxz - Sqrt[3]*Eappyz - (2*I)*Sqrt[3]*Mappyzxz - 4*Mapx2y2z2 + (4*I)*Mapz2xy)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Eapx2y2 - Eapxy + (2*I)*Mapx2y2xy))/2, k == 4 && m == -4}, {(3*(Eappxz - Eappyz + (2*I)*Mappyzxz + Sqrt[3]*Mapx2y2z2 + I*Sqrt[3]*Mapz2xy))/Sqrt[10], k == 4 && m == -2}, {(-3*(4*Eappxz + 4*Eappyz - Eapx2y2 - Eapxy - 6*Eapz2))/10, k == 4 && m == 0}, {(3*(Eappxz - Eappyz - (2*I)*Mappyzxz + Sqrt[3]*Mapx2y2z2 - I*Sqrt[3]*Mapz2xy))/Sqrt[10], k == 4 && m == 2}}, (3*Sqrt[7/10]*(Eapx2y2 - Eapxy - (2*I)*Mapx2y2xy))/2] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{0, 0, (1/5)*(Eappxz + Eappyz + Eapx2y2 + Eapxy + Eapz2)} ,  
 +       {2, 0, (1/2)*(Eappxz + Eappyz + (-2)*(Eapx2y2 + Eapxy + (-1)*(Eapz2)))} ,  
 +       {2, 2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Eappxz) + (-1)*((sqrt(3))*(Eappyz)) + (-2*I)*((sqrt(3))*(Mappyzxz)) + (-4)*(Mapx2y2z2) + (4*I)*(Mapz2xy)))} ,  
 +       {2,-2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Eappxz) + (-1)*((sqrt(3))*(Eappyz)) + (2*I)*((sqrt(3))*(Mappyzxz)) + (-4)*(Mapx2y2z2) + (-4*I)*(Mapz2xy)))} ,  
 +       {4, 0, (-3/10)*((4)*(Eappxz) + (4)*(Eappyz) + (-1)*(Eapx2y2) + (-1)*(Eapxy) + (-6)*(Eapz2))} ,  
 +       {4, 2, (3)*((1/(sqrt(10)))*(Eappxz + (-1)*(Eappyz) + (-2*I)*(Mappyzxz) + (sqrt(3))*(Mapx2y2z2) + (-I)*((sqrt(3))*(Mapz2xy))))} ,  
 +       {4,-2, (3)*((1/(sqrt(10)))*(Eappxz + (-1)*(Eappyz) + (2*I)*(Mappyzxz) + (sqrt(3))*(Mapx2y2z2) + (I)*((sqrt(3))*(Mapz2xy))))} ,  
 +       {4, 4, (3/2)*((sqrt(7/10))*(Eapx2y2 + (-1)*(Eapxy) + (-2*I)*(Mapx2y2xy)))} ,  
 +       {4,-4, (3/2)*((sqrt(7/10))*(Eapx2y2 + (-1)*(Eapxy) + (2*I)*(Mapx2y2xy)))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** > 
 + 
 +### 
 + 
 +|      ^  {Y_{-2}^{(2)}}   ^  {Y_{-1}^{(2)}}   ^  {Y_{0}^{(2)}}   ^  {Y_{1}^{(2)}}   ^  {Y_{2}^{(2)}}   ^ 
 +^ {Y_{-2}^{(2)}} | \frac{\text{Eapx2y2}+\text{Eapxy}}{2} | 0 | \frac{\text{Mapx2y2z2}+i \text{Mapz2xy}}{\sqrt{2}} | 0 | \frac{1}{2} (\text{Eapx2y2}-\text{Eapxy}+2 i \text{Mapx2y2xy}) | 
 +^ {Y_{-1}^{(2)}} | 0 | \frac{\text{Eappxz}+\text{Eappyz}}{2} | 0 | \frac{1}{2} (-\text{Eappxz}+\text{Eappyz}-2 i \text{Mappyzxz}) | 0 | 
 +^ {Y_{0}^{(2)}} | \frac{\text{Mapx2y2z2}-i \text{Mapz2xy}}{\sqrt{2}} | 0 | \text{Eapz2} | 0 | \frac{\text{Mapx2y2z2}+i \text{Mapz2xy}}{\sqrt{2}} | 
 +^ {Y_{1}^{(2)}} | 0 | \frac{1}{2} (-\text{Eappxz}+\text{Eappyz}+2 i \text{Mappyzxz}) | 0 | \frac{\text{Eappxz}+\text{Eappyz}}{2} | 0 | 
 +^ {Y_{2}^{(2)}} | \frac{1}{2} (\text{Eapx2y2}-\text{Eapxy}-2 i \text{Mapx2y2xy}) | 0 | \frac{\text{Mapx2y2z2}-i \text{Mapz2xy}}{\sqrt{2}} | 0 | \frac{\text{Eapx2y2}+\text{Eapxy}}{2} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  d_{x^2-y^2}   ^  d_{3z^2-r^2}   ^  d_{\text{yz}}   ^  d_{\text{xz}}   ^  d_{\text{xy}}   ^ 
 +^ d_{x^2-y^2} | \text{Eapx2y2} | \text{Mapx2y2z2} | 0 | 0 | \text{Mapx2y2xy} | 
 +^ d_{3z^2-r^2} | \text{Mapx2y2z2} | \text{Eapz2} | 0 | 0 | \text{Mapz2xy} | 
 +^ d_{\text{yz}} | 0 | 0 | \text{Eappyz} | \text{Mappyzxz} | 0 | 
 +^ d_{\text{xz}} | 0 | 0 | \text{Mappyzxz} | \text{Eappxz} | 0 | 
 +^ d_{\text{xy}} | \text{Mapx2y2xy} | \text{Mapz2xy} | 0 | 0 | \text{Eapxy} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Rotation matrix used** > 
 + 
 +### 
 + 
 +|      ^  {Y_{-2}^{(2)}}   ^  {Y_{-1}^{(2)}}   ^  {Y_{0}^{(2)}}   ^  {Y_{1}^{(2)}}   ^  {Y_{2}^{(2)}}   ^ 
 +^ d_{x^2-y^2} | \frac{1}{\sqrt{2}} | 0 | 0 | 0 | \frac{1}{\sqrt{2}} | 
 +^ d_{3z^2-r^2} | 0 | 0 | 1 | 0 | 0 | 
 +^ d_{\text{yz}} | 0 | \frac{i}{\sqrt{2}} | 0 | \frac{i}{\sqrt{2}} | 0 | 
 +^ d_{\text{xz}} | 0 | \frac{1}{\sqrt{2}} | 0 | -\frac{1}{\sqrt{2}} | 0 | 
 +^ d_{\text{xy}} | \frac{i}{\sqrt{2}} | 0 | 0 | 0 | -\frac{i}{\sqrt{2}} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Irriducible representations and their onsite energy** > 
 + 
 +### 
 + 
 +^ ^\text{Eapx2y2} | {{:physics_chemistry:pointgroup:cs_z_orb_2_1.png?150}} | 
 +|\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi ) | ::: | 
 +|\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right) | ::: | 
 +^ ^\text{Eapz2} | {{:physics_chemistry:pointgroup:cs_z_orb_2_2.png?150}} | 
 +|\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1) | ::: | 
 +|\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right) | ::: | 
 +^ ^\text{Eappyz} | {{:physics_chemistry:pointgroup:cs_z_orb_2_3.png?150}} | 
 +|\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi ) | ::: | 
 +|\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |\frac{1}{2} \sqrt{\frac{15}{\pi }} y z | ::: | 
 +^ ^\text{Eappxz} | {{:physics_chemistry:pointgroup:cs_z_orb_2_4.png?150}} | 
 +|\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{Eapxy} | {{:physics_chemistry:pointgroup:cs_z_orb_2_5.png?150}} | 
 +|\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 | ::: | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for f orbitals ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + \frac{1}{7} (\text{Eappx3}+\text{Eappxy2z2}+\text{Eappxyz}+\text{Eappy3}+\text{Eappyz2x2}+\text{Eappz3}+\text{Eappzx2y2}) & k=0\land m=0 \\ 
 + 0 & (k\neq 6\land (((k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2))\land k\neq 4)\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4)))\lor (m\neq -6\land m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4\land m\neq 6) \\ 
 + \frac{5 \left(2 \sqrt{3} \text{Eappx3}-2 \sqrt{3} \text{Eappy3}+2 \sqrt{5} \text{Mappx3xy2z2}-i \sqrt{3} \text{Mappx3y3}-i \sqrt{5} \text{Mappx3yz2x2}-5 i \sqrt{3} \text{Mappxy2z2yz2x2}-4 i \sqrt{5} \text{Mappxyzz3}+i \sqrt{5} \text{Mappy3xy2z2}+2 \sqrt{5} \text{Mappy3yz2x2}-4 \sqrt{5} \text{Mappz3zx2y2}\right)}{28 \sqrt{2}} & k=2\land m=-2 \\ 
 + -\frac{5}{14} \left(\text{Eappx3}+\text{Eappy3}-2 \text{Eappz3}-\sqrt{15} \text{Mappx3xy2z2}+\sqrt{15} \text{Mappy3yz2x2}\right) & k=2\land m=0 \\ 
 + \frac{5 \left(2 \sqrt{3} \text{Eappx3}-2 \sqrt{3} \text{Eappy3}+2 \sqrt{5} \text{Mappx3xy2z2}+i \sqrt{3} \text{Mappx3y3}+i \sqrt{5} \text{Mappx3yz2x2}+5 i \sqrt{3} \text{Mappxy2z2yz2x2}+4 i \sqrt{5} \text{Mappxyzz3}-i \sqrt{5} \text{Mappy3xy2z2}+2 \sqrt{5} \text{Mappy3yz2x2}-4 \sqrt{5} \text{Mappz3zx2y2}\right)}{28 \sqrt{2}} & k=2\land m=2 \\ 
 + \frac{3 \left(3 \sqrt{5} \text{Eappx3}-3 \sqrt{5} \text{Eappxy2z2}-4 \sqrt{5} \text{Eappxyz}+3 \sqrt{5} \text{Eappy3}-3 \sqrt{5} \text{Eappyz2x2}+4 \sqrt{5} \text{Eappzx2y2}+2 \sqrt{3} \text{Mappx3xy2z2}-8 i \sqrt{3} \text{Mappx3yz2x2}+8 i \sqrt{5} \text{Mappxyzzx2y2}-8 i \sqrt{3} \text{Mappy3xy2z2}-2 \sqrt{3} \text{Mappy3yz2x2}\right)}{8 \sqrt{14}} & k=4\land m=-4 \\ 
 + \frac{3}{56} \left(-3 \sqrt{10} \text{Eappx3}+7 \sqrt{10} \text{Eappxy2z2}+3 \sqrt{10} \text{Eappy3}-7 \sqrt{10} \text{Eappyz2x2}+2 \sqrt{6} \text{Mappx3xy2z2}+12 i \sqrt{10} \text{Mappx3y3}-8 i \sqrt{6} \text{Mappx3yz2x2}+4 i \sqrt{10} \text{Mappxy2z2yz2x2}-4 i \sqrt{6} \text{Mappxyzz3}+8 i \sqrt{6} \text{Mappy3xy2z2}+2 \sqrt{6} \text{Mappy3yz2x2}-4 \sqrt{6} \text{Mappz3zx2y2}\right) & k=4\land m=-2 \\ 
 + \frac{3}{56} \left(9 \text{Eappx3}+7 \text{Eappxy2z2}-28 \text{Eappxyz}+9 \text{Eappy3}+7 \text{Eappyz2x2}+24 \text{Eappz3}-28 \text{Eappzx2y2}-2 \sqrt{15} \text{Mappx3xy2z2}+2 \sqrt{15} \text{Mappy3yz2x2}\right) & k=4\land m=0 \\ 
 + \frac{3}{56} \left(-3 \sqrt{10} \text{Eappx3}+7 \sqrt{10} \text{Eappxy2z2}+3 \sqrt{10} \text{Eappy3}-7 \sqrt{10} \text{Eappyz2x2}+2 \sqrt{6} \text{Mappx3xy2z2}-12 i \sqrt{10} \text{Mappx3y3}+8 i \sqrt{6} \text{Mappx3yz2x2}-4 i \sqrt{10} \text{Mappxy2z2yz2x2}+4 i \sqrt{6} \text{Mappxyzz3}-8 i \sqrt{6} \text{Mappy3xy2z2}+2 \sqrt{6} \text{Mappy3yz2x2}-4 \sqrt{6} \text{Mappz3zx2y2}\right) & k=4\land m=2 \\ 
 + \frac{3 \left(3 \sqrt{5} \text{Eappx3}-3 \sqrt{5} \text{Eappxy2z2}-4 \sqrt{5} \text{Eappxyz}+3 \sqrt{5} \text{Eappy3}-3 \sqrt{5} \text{Eappyz2x2}+4 \sqrt{5} \text{Eappzx2y2}+2 \sqrt{3} \text{Mappx3xy2z2}+8 i \sqrt{3} \text{Mappx3yz2x2}-8 i \sqrt{5} \text{Mappxyzzx2y2}+8 i \sqrt{3} \text{Mappy3xy2z2}-2 \sqrt{3} \text{Mappy3yz2x2}\right)}{8 \sqrt{14}} & k=4\land m=4 \\ 
 + \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappx3}+3 \sqrt{3} \text{Eappxy2z2}-5 \sqrt{3} \text{Eappy3}-3 \sqrt{3} \text{Eappyz2x2}-6 \sqrt{5} \text{Mappx3xy2z2}-10 i \sqrt{3} \text{Mappx3y3}-6 i \sqrt{5} \text{Mappx3yz2x2}+6 i \sqrt{3} \text{Mappxy2z2yz2x2}+6 i \sqrt{5} \text{Mappy3xy2z2}-6 \sqrt{5} \text{Mappy3yz2x2}\right) & k=6\land m=-6 \\ 
 + -\frac{13 \left(15 \text{Eappx3}-15 \text{Eappxy2z2}+24 \text{Eappxyz}+15 \text{Eappy3}-15 \text{Eappyz2x2}-24 \text{Eappzx2y2}+2 \sqrt{15} \text{Mappx3xy2z2}-8 i \sqrt{15} \text{Mappx3yz2x2}-48 i \text{Mappxyzzx2y2}-8 i \sqrt{15} \text{Mappy3xy2z2}-2 \sqrt{15} \text{Mappy3yz2x2}\right)}{80 \sqrt{14}} & k=6\land m=-4 \\ 
 + \frac{13 \left(5 \sqrt{15} \text{Eappx3}+3 \sqrt{15} \text{Eappxy2z2}-5 \sqrt{15} \text{Eappy3}-3 \sqrt{15} \text{Eappyz2x2}+34 \text{Mappx3xy2z2}+2 i \sqrt{15} \text{Mappx3y3}-26 i \text{Mappx3yz2x2}-14 i \sqrt{15} \text{Mappxy2z2yz2x2}+64 i \text{Mappxyzz3}+26 i \text{Mappy3xy2z2}+34 \text{Mappy3yz2x2}+64 \text{Mappz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=-2 \\ 
 + -\frac{13}{560} \left(25 \text{Eappx3}+39 \text{Eappxy2z2}-24 \text{Eappxyz}+25 \text{Eappy3}+39 \text{Eappyz2x2}-80 \text{Eappz3}-24 \text{Eappzx2y2}+14 \sqrt{15} \text{Mappx3xy2z2}-14 \sqrt{15} \text{Mappy3yz2x2}\right) & k=6\land m=0 \\ 
 + \frac{13 \left(5 \sqrt{15} \text{Eappx3}+3 \sqrt{15} \text{Eappxy2z2}-5 \sqrt{15} \text{Eappy3}-3 \sqrt{15} \text{Eappyz2x2}+34 \text{Mappx3xy2z2}-2 i \sqrt{15} \text{Mappx3y3}+26 i \text{Mappx3yz2x2}+14 i \sqrt{15} \text{Mappxy2z2yz2x2}-64 i \text{Mappxyzz3}-26 i \text{Mappy3xy2z2}+34 \text{Mappy3yz2x2}+64 \text{Mappz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=2 \\ 
 + -\frac{13 \left(15 \text{Eappx3}-15 \text{Eappxy2z2}+24 \text{Eappxyz}+15 \text{Eappy3}-15 \text{Eappyz2x2}-24 \text{Eappzx2y2}+2 \sqrt{15} \text{Mappx3xy2z2}+8 i \sqrt{15} \text{Mappx3yz2x2}+48 i \text{Mappxyzzx2y2}+8 i \sqrt{15} \text{Mappy3xy2z2}-2 \sqrt{15} \text{Mappy3yz2x2}\right)}{80 \sqrt{14}} & k=6\land m=4 \\ 
 + \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappx3}+3 \sqrt{3} \text{Eappxy2z2}-5 \sqrt{3} \text{Eappy3}-3 \sqrt{3} \text{Eappyz2x2}-6 \sqrt{5} \text{Mappx3xy2z2}+10 i \sqrt{3} \text{Mappx3y3}+6 i \sqrt{5} \text{Mappx3yz2x2}-6 i \sqrt{3} \text{Mappxy2z2yz2x2}-6 i \sqrt{5} \text{Mappy3xy2z2}-6 \sqrt{5} \text{Mappy3yz2x2}\right) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{(Eappx3 + Eappxy2z2 + Eappxyz + Eappy3 + Eappyz2x2 + Eappz3 + Eappzx2y2)/7, k == 0 && m == 0}, {0, (k != 6 && (((k != 2 || (m != -2 && m != 0 && m != 2)) && k != 4) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4))) || (m != -6 && m != -4 && m != -2 && m != 0 && m != 2 && m != 4 && m != 6)}, {(5*(2*Sqrt[3]*Eappx3 - 2*Sqrt[3]*Eappy3 + 2*Sqrt[5]*Mappx3xy2z2 - I*Sqrt[3]*Mappx3y3 - I*Sqrt[5]*Mappx3yz2x2 - (5*I)*Sqrt[3]*Mappxy2z2yz2x2 - (4*I)*Sqrt[5]*Mappxyzz3 + I*Sqrt[5]*Mappy3xy2z2 + 2*Sqrt[5]*Mappy3yz2x2 - 4*Sqrt[5]*Mappz3zx2y2))/(28*Sqrt[2]), k == 2 && m == -2}, {(-5*(Eappx3 + Eappy3 - 2*Eappz3 - Sqrt[15]*Mappx3xy2z2 + Sqrt[15]*Mappy3yz2x2))/14, k == 2 && m == 0}, {(5*(2*Sqrt[3]*Eappx3 - 2*Sqrt[3]*Eappy3 + 2*Sqrt[5]*Mappx3xy2z2 + I*Sqrt[3]*Mappx3y3 + I*Sqrt[5]*Mappx3yz2x2 + (5*I)*Sqrt[3]*Mappxy2z2yz2x2 + (4*I)*Sqrt[5]*Mappxyzz3 - I*Sqrt[5]*Mappy3xy2z2 + 2*Sqrt[5]*Mappy3yz2x2 - 4*Sqrt[5]*Mappz3zx2y2))/(28*Sqrt[2]), k == 2 && m == 2}, {(3*(3*Sqrt[5]*Eappx3 - 3*Sqrt[5]*Eappxy2z2 - 4*Sqrt[5]*Eappxyz + 3*Sqrt[5]*Eappy3 - 3*Sqrt[5]*Eappyz2x2 + 4*Sqrt[5]*Eappzx2y2 + 2*Sqrt[3]*Mappx3xy2z2 - (8*I)*Sqrt[3]*Mappx3yz2x2 + (8*I)*Sqrt[5]*Mappxyzzx2y2 - (8*I)*Sqrt[3]*Mappy3xy2z2 - 2*Sqrt[3]*Mappy3yz2x2))/(8*Sqrt[14]), k == 4 && m == -4}, {(3*(-3*Sqrt[10]*Eappx3 + 7*Sqrt[10]*Eappxy2z2 + 3*Sqrt[10]*Eappy3 - 7*Sqrt[10]*Eappyz2x2 + 2*Sqrt[6]*Mappx3xy2z2 + (12*I)*Sqrt[10]*Mappx3y3 - (8*I)*Sqrt[6]*Mappx3yz2x2 + (4*I)*Sqrt[10]*Mappxy2z2yz2x2 - (4*I)*Sqrt[6]*Mappxyzz3 + (8*I)*Sqrt[6]*Mappy3xy2z2 + 2*Sqrt[6]*Mappy3yz2x2 - 4*Sqrt[6]*Mappz3zx2y2))/56, k == 4 && m == -2}, {(3*(9*Eappx3 + 7*Eappxy2z2 - 28*Eappxyz + 9*Eappy3 + 7*Eappyz2x2 + 24*Eappz3 - 28*Eappzx2y2 - 2*Sqrt[15]*Mappx3xy2z2 + 2*Sqrt[15]*Mappy3yz2x2))/56, k == 4 && m == 0}, {(3*(-3*Sqrt[10]*Eappx3 + 7*Sqrt[10]*Eappxy2z2 + 3*Sqrt[10]*Eappy3 - 7*Sqrt[10]*Eappyz2x2 + 2*Sqrt[6]*Mappx3xy2z2 - (12*I)*Sqrt[10]*Mappx3y3 + (8*I)*Sqrt[6]*Mappx3yz2x2 - (4*I)*Sqrt[10]*Mappxy2z2yz2x2 + (4*I)*Sqrt[6]*Mappxyzz3 - (8*I)*Sqrt[6]*Mappy3xy2z2 + 2*Sqrt[6]*Mappy3yz2x2 - 4*Sqrt[6]*Mappz3zx2y2))/56, k == 4 && m == 2}, {(3*(3*Sqrt[5]*Eappx3 - 3*Sqrt[5]*Eappxy2z2 - 4*Sqrt[5]*Eappxyz + 3*Sqrt[5]*Eappy3 - 3*Sqrt[5]*Eappyz2x2 + 4*Sqrt[5]*Eappzx2y2 + 2*Sqrt[3]*Mappx3xy2z2 + (8*I)*Sqrt[3]*Mappx3yz2x2 - (8*I)*Sqrt[5]*Mappxyzzx2y2 + (8*I)*Sqrt[3]*Mappy3xy2z2 - 2*Sqrt[3]*Mappy3yz2x2))/(8*Sqrt[14]), k == 4 && m == 4}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eappx3 + 3*Sqrt[3]*Eappxy2z2 - 5*Sqrt[3]*Eappy3 - 3*Sqrt[3]*Eappyz2x2 - 6*Sqrt[5]*Mappx3xy2z2 - (10*I)*Sqrt[3]*Mappx3y3 - (6*I)*Sqrt[5]*Mappx3yz2x2 + (6*I)*Sqrt[3]*Mappxy2z2yz2x2 + (6*I)*Sqrt[5]*Mappy3xy2z2 - 6*Sqrt[5]*Mappy3yz2x2))/160, k == 6 && m == -6}, {(-13*(15*Eappx3 - 15*Eappxy2z2 + 24*Eappxyz + 15*Eappy3 - 15*Eappyz2x2 - 24*Eappzx2y2 + 2*Sqrt[15]*Mappx3xy2z2 - (8*I)*Sqrt[15]*Mappx3yz2x2 - (48*I)*Mappxyzzx2y2 - (8*I)*Sqrt[15]*Mappy3xy2z2 - 2*Sqrt[15]*Mappy3yz2x2))/(80*Sqrt[14]), k == 6 && m == -4}, {(13*(5*Sqrt[15]*Eappx3 + 3*Sqrt[15]*Eappxy2z2 - 5*Sqrt[15]*Eappy3 - 3*Sqrt[15]*Eappyz2x2 + 34*Mappx3xy2z2 + (2*I)*Sqrt[15]*Mappx3y3 - (26*I)*Mappx3yz2x2 - (14*I)*Sqrt[15]*Mappxy2z2yz2x2 + (64*I)*Mappxyzz3 + (26*I)*Mappy3xy2z2 + 34*Mappy3yz2x2 + 64*Mappz3zx2y2))/(160*Sqrt[7]), k == 6 && m == -2}, {(-13*(25*Eappx3 + 39*Eappxy2z2 - 24*Eappxyz + 25*Eappy3 + 39*Eappyz2x2 - 80*Eappz3 - 24*Eappzx2y2 + 14*Sqrt[15]*Mappx3xy2z2 - 14*Sqrt[15]*Mappy3yz2x2))/560, k == 6 && m == 0}, {(13*(5*Sqrt[15]*Eappx3 + 3*Sqrt[15]*Eappxy2z2 - 5*Sqrt[15]*Eappy3 - 3*Sqrt[15]*Eappyz2x2 + 34*Mappx3xy2z2 - (2*I)*Sqrt[15]*Mappx3y3 + (26*I)*Mappx3yz2x2 + (14*I)*Sqrt[15]*Mappxy2z2yz2x2 - (64*I)*Mappxyzz3 - (26*I)*Mappy3xy2z2 + 34*Mappy3yz2x2 + 64*Mappz3zx2y2))/(160*Sqrt[7]), k == 6 && m == 2}, {(-13*(15*Eappx3 - 15*Eappxy2z2 + 24*Eappxyz + 15*Eappy3 - 15*Eappyz2x2 - 24*Eappzx2y2 + 2*Sqrt[15]*Mappx3xy2z2 + (8*I)*Sqrt[15]*Mappx3yz2x2 + (48*I)*Mappxyzzx2y2 + (8*I)*Sqrt[15]*Mappy3xy2z2 - 2*Sqrt[15]*Mappy3yz2x2))/(80*Sqrt[14]), k == 6 && m == 4}}, (13*Sqrt[11/7]*(5*Sqrt[3]*Eappx3 + 3*Sqrt[3]*Eappxy2z2 - 5*Sqrt[3]*Eappy3 - 3*Sqrt[3]*Eappyz2x2 - 6*Sqrt[5]*Mappx3xy2z2 + (10*I)*Sqrt[3]*Mappx3y3 + (6*I)*Sqrt[5]*Mappx3yz2x2 - (6*I)*Sqrt[3]*Mappxy2z2yz2x2 - (6*I)*Sqrt[5]*Mappy3xy2z2 - 6*Sqrt[5]*Mappy3yz2x2))/160] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{0, 0, (1/7)*(Eappx3 + Eappxy2z2 + Eappxyz + Eappy3 + Eappyz2x2 + Eappz3 + Eappzx2y2)} ,  
 +       {2, 0, (-5/14)*(Eappx3 + Eappy3 + (-2)*(Eappz3) + (-1)*((sqrt(15))*(Mappx3xy2z2)) + (sqrt(15))*(Mappy3yz2x2))} ,  
 +       {2,-2, (5/28)*((1/(sqrt(2)))*((2)*((sqrt(3))*(Eappx3)) + (-2)*((sqrt(3))*(Eappy3)) + (2)*((sqrt(5))*(Mappx3xy2z2)) + (-I)*((sqrt(3))*(Mappx3y3)) + (-I)*((sqrt(5))*(Mappx3yz2x2)) + (-5*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (-4*I)*((sqrt(5))*(Mappxyzz3)) + (I)*((sqrt(5))*(Mappy3xy2z2)) + (2)*((sqrt(5))*(Mappy3yz2x2)) + (-4)*((sqrt(5))*(Mappz3zx2y2))))} ,  
 +       {2, 2, (5/28)*((1/(sqrt(2)))*((2)*((sqrt(3))*(Eappx3)) + (-2)*((sqrt(3))*(Eappy3)) + (2)*((sqrt(5))*(Mappx3xy2z2)) + (I)*((sqrt(3))*(Mappx3y3)) + (I)*((sqrt(5))*(Mappx3yz2x2)) + (5*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (4*I)*((sqrt(5))*(Mappxyzz3)) + (-I)*((sqrt(5))*(Mappy3xy2z2)) + (2)*((sqrt(5))*(Mappy3yz2x2)) + (-4)*((sqrt(5))*(Mappz3zx2y2))))} ,  
 +       {4, 0, (3/56)*((9)*(Eappx3) + (7)*(Eappxy2z2) + (-28)*(Eappxyz) + (9)*(Eappy3) + (7)*(Eappyz2x2) + (24)*(Eappz3) + (-28)*(Eappzx2y2) + (-2)*((sqrt(15))*(Mappx3xy2z2)) + (2)*((sqrt(15))*(Mappy3yz2x2)))} ,  
 +       {4, 2, (3/56)*((-3)*((sqrt(10))*(Eappx3)) + (7)*((sqrt(10))*(Eappxy2z2)) + (3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (2)*((sqrt(6))*(Mappx3xy2z2)) + (-12*I)*((sqrt(10))*(Mappx3y3)) + (8*I)*((sqrt(6))*(Mappx3yz2x2)) + (-4*I)*((sqrt(10))*(Mappxy2z2yz2x2)) + (4*I)*((sqrt(6))*(Mappxyzz3)) + (-8*I)*((sqrt(6))*(Mappy3xy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (-4)*((sqrt(6))*(Mappz3zx2y2)))} ,  
 +       {4,-2, (3/56)*((-3)*((sqrt(10))*(Eappx3)) + (7)*((sqrt(10))*(Eappxy2z2)) + (3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (2)*((sqrt(6))*(Mappx3xy2z2)) + (12*I)*((sqrt(10))*(Mappx3y3)) + (-8*I)*((sqrt(6))*(Mappx3yz2x2)) + (4*I)*((sqrt(10))*(Mappxy2z2yz2x2)) + (-4*I)*((sqrt(6))*(Mappxyzz3)) + (8*I)*((sqrt(6))*(Mappy3xy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (-4)*((sqrt(6))*(Mappz3zx2y2)))} ,  
 +       {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eappy3)) + (-3)*((sqrt(5))*(Eappyz2x2)) + (4)*((sqrt(5))*(Eappzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (-8*I)*((sqrt(3))*(Mappx3yz2x2)) + (8*I)*((sqrt(5))*(Mappxyzzx2y2)) + (-8*I)*((sqrt(3))*(Mappy3xy2z2)) + (-2)*((sqrt(3))*(Mappy3yz2x2))))} ,  
 +       {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eappx3)) + (-3)*((sqrt(5))*(Eappxy2z2)) + (-4)*((sqrt(5))*(Eappxyz)) + (3)*((sqrt(5))*(Eappy3)) + (-3)*((sqrt(5))*(Eappyz2x2)) + (4)*((sqrt(5))*(Eappzx2y2)) + (2)*((sqrt(3))*(Mappx3xy2z2)) + (8*I)*((sqrt(3))*(Mappx3yz2x2)) + (-8*I)*((sqrt(5))*(Mappxyzzx2y2)) + (8*I)*((sqrt(3))*(Mappy3xy2z2)) + (-2)*((sqrt(3))*(Mappy3yz2x2))))} ,  
 +       {6, 0, (-13/560)*((25)*(Eappx3) + (39)*(Eappxy2z2) + (-24)*(Eappxyz) + (25)*(Eappy3) + (39)*(Eappyz2x2) + (-80)*(Eappz3) + (-24)*(Eappzx2y2) + (14)*((sqrt(15))*(Mappx3xy2z2)) + (-14)*((sqrt(15))*(Mappy3yz2x2)))} ,  
 +       {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eappy3)) + (-3)*((sqrt(15))*(Eappyz2x2)) + (34)*(Mappx3xy2z2) + (-2*I)*((sqrt(15))*(Mappx3y3)) + (26*I)*(Mappx3yz2x2) + (14*I)*((sqrt(15))*(Mappxy2z2yz2x2)) + (-64*I)*(Mappxyzz3) + (-26*I)*(Mappy3xy2z2) + (34)*(Mappy3yz2x2) + (64)*(Mappz3zx2y2)))} ,  
 +       {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappx3)) + (3)*((sqrt(15))*(Eappxy2z2)) + (-5)*((sqrt(15))*(Eappy3)) + (-3)*((sqrt(15))*(Eappyz2x2)) + (34)*(Mappx3xy2z2) + (2*I)*((sqrt(15))*(Mappx3y3)) + (-26*I)*(Mappx3yz2x2) + (-14*I)*((sqrt(15))*(Mappxy2z2yz2x2)) + (64*I)*(Mappxyzz3) + (26*I)*(Mappy3xy2z2) + (34)*(Mappy3yz2x2) + (64)*(Mappz3zx2y2)))} ,  
 +       {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (-24)*(Eappzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (-8*I)*((sqrt(15))*(Mappx3yz2x2)) + (-48*I)*(Mappxyzzx2y2) + (-8*I)*((sqrt(15))*(Mappy3xy2z2)) + (-2)*((sqrt(15))*(Mappy3yz2x2))))} ,  
 +       {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Eappx3) + (-15)*(Eappxy2z2) + (24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (-24)*(Eappzx2y2) + (2)*((sqrt(15))*(Mappx3xy2z2)) + (8*I)*((sqrt(15))*(Mappx3yz2x2)) + (48*I)*(Mappxyzzx2y2) + (8*I)*((sqrt(15))*(Mappy3xy2z2)) + (-2)*((sqrt(15))*(Mappy3yz2x2))))} ,  
 +       {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eappy3)) + (-3)*((sqrt(3))*(Eappyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (-10*I)*((sqrt(3))*(Mappx3y3)) + (-6*I)*((sqrt(5))*(Mappx3yz2x2)) + (6*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (6*I)*((sqrt(5))*(Mappy3xy2z2)) + (-6)*((sqrt(5))*(Mappy3yz2x2))))} ,  
 +       {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappx3)) + (3)*((sqrt(3))*(Eappxy2z2)) + (-5)*((sqrt(3))*(Eappy3)) + (-3)*((sqrt(3))*(Eappyz2x2)) + (-6)*((sqrt(5))*(Mappx3xy2z2)) + (10*I)*((sqrt(3))*(Mappx3y3)) + (6*I)*((sqrt(5))*(Mappx3yz2x2)) + (-6*I)*((sqrt(3))*(Mappxy2z2yz2x2)) + (-6*I)*((sqrt(5))*(Mappy3xy2z2)) + (-6)*((sqrt(5))*(Mappy3yz2x2))))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **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{Eappy3}+3 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mappx3xy2z2})\right) | 0 | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}+2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}+i \text{Mappy3yz2x2}\right)\right) | 0 | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}-4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \left(\sqrt{15} \text{Mappx3xy2z2}+5 i \text{Mappx3y3}+i \sqrt{15} \text{Mappx3yz2x2}-3 i \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3yz2x2}-i \text{Mappy3xy2z2})\right)\right) | 
 +^ {Y_{-2}^{(3)}} | 0 | \frac{\text{Eappxyz}+\text{Eappzx2y2}}{2} | 0 | \frac{\text{Mappz3zx2y2}+i \text{Mappxyzz3}}{\sqrt{2}} | 0 | \frac{1}{2} (-\text{Eappxyz}+\text{Eappzx2y2}+2 i \text{Mappxyzzx2y2}) | 0 | 
 +^ {Y_{-1}^{(3)}} | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}-2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}-i \text{Mappy3yz2x2}\right)\right) | 0 | \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}-2 \left(\sqrt{15} \text{Mappx3xy2z2}+3 i \text{Mappx3y3}-i \sqrt{15} \text{Mappx3yz2x2}-5 i \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3yz2x2}+i \text{Mappy3xy2z2})\right)\right) | 0 | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}-4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 
 +^ {Y_{0}^{(3)}} | 0 | \frac{\text{Mappz3zx2y2}-i \text{Mappxyzz3}}{\sqrt{2}} | 0 | \text{Eappz3} | 0 | \frac{\text{Mappz3zx2y2}+i \text{Mappxyzz3}}{\sqrt{2}} | 0 | 
 +^ {Y_{1}^{(3)}} | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}+4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-3 \text{Eappx3}-5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}-2 \sqrt{15} \text{Mappx3xy2z2}+2 i \left(3 \text{Mappx3y3}-\sqrt{15} \text{Mappx3yz2x2}-5 \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3xy2z2}+i \text{Mappy3yz2x2})\right)\right) | 0 | \frac{1}{16} \left(3 \text{Eappx3}+5 \text{Eappxy2z2}+3 \text{Eappy3}+5 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappx3xy2z2}-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}+2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}+i \text{Mappy3yz2x2}\right)\right) | 
 +^ {Y_{2}^{(3)}} | 0 | \frac{1}{2} (-\text{Eappxyz}+\text{Eappzx2y2}-2 i \text{Mappxyzzx2y2}) | 0 | \frac{\text{Mappz3zx2y2}-i \text{Mappxyzz3}}{\sqrt{2}} | 0 | \frac{\text{Eappxyz}+\text{Eappzx2y2}}{2} | 0 | 
 +^ {Y_{3}^{(3)}} | \frac{1}{16} \left(-5 \text{Eappx3}-3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \left(\sqrt{15} \text{Mappx3xy2z2}-5 i \text{Mappx3y3}-i \sqrt{15} \text{Mappx3yz2x2}+3 i \text{Mappxy2z2yz2x2}+\sqrt{15} (\text{Mappy3yz2x2}+i \text{Mappy3xy2z2})\right)\right) | 0 | \frac{1}{16} \left(\sqrt{15} \text{Eappx3}-\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+2 (\text{Mappx3xy2z2}+4 i (\text{Mappx3yz2x2}+\text{Mappy3xy2z2})-\text{Mappy3yz2x2})\right) | 0 | \frac{1}{16} \left(-\sqrt{15} \text{Eappx3}+\sqrt{15} \text{Eappxy2z2}+\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-2 \text{Mappx3xy2z2}-2 i \left(\sqrt{15} \text{Mappx3y3}-\text{Mappx3yz2x2}+\sqrt{15} \text{Mappxy2z2yz2x2}+\text{Mappy3xy2z2}-i \text{Mappy3yz2x2}\right)\right) | 0 | \frac{1}{16} \left(5 \text{Eappx3}+3 \text{Eappxy2z2}+5 \text{Eappy3}+3 \text{Eappyz2x2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mappx3xy2z2})\right) | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  f_{\text{xyz}}   ^  f_{x\left(5x^2-r^2\right)}   ^  f_{y\left(5y^2-r^2\right)}   ^  f_{z\left(5z^2-r^2\right)}   ^  f_{x\left(y^2-z^2\right)}   ^  f_{y\left(z^2-x^2\right)}   ^  f_{z\left(x^2-y^2\right)}   ^ 
 +^ f_{\text{xyz}} | \text{Eappxyz} | 0 | 0 | \text{Mappxyzz3} | 0 | 0 | \text{Mappxyzzx2y2} | 
 +^ f_{x\left(5x^2-r^2\right)} | 0 | \text{Eappx3} | \text{Mappx3y3} | 0 | \text{Mappx3xy2z2} | \text{Mappx3yz2x2} | 0 | 
 +^ f_{y\left(5y^2-r^2\right)} | 0 | \text{Mappx3y3} | \text{Eappy3} | 0 | \text{Mappy3xy2z2} | \text{Mappy3yz2x2} | 0 | 
 +^ f_{z\left(5z^2-r^2\right)} | \text{Mappxyzz3} | 0 | 0 | \text{Eappz3} | 0 | 0 | \text{Mappz3zx2y2} | 
 +^ f_{x\left(y^2-z^2\right)} | 0 | \text{Mappx3xy2z2} | \text{Mappy3xy2z2} | 0 | \text{Eappxy2z2} | \text{Mappxy2z2yz2x2} | 0 | 
 +^ f_{y\left(z^2-x^2\right)} | 0 | \text{Mappx3yz2x2} | \text{Mappy3yz2x2} | 0 | \text{Mappxy2z2yz2x2} | \text{Eappyz2x2} | 0 | 
 +^ f_{z\left(x^2-y^2\right)} | \text{Mappxyzzx2y2} | 0 | 0 | \text{Mappz3zx2y2} | 0 | 0 | \text{Eappzx2y2} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Rotation matrix used** > 
 + 
 +### 
 + 
 +|      ^  {Y_{-3}^{(3)}}   ^  {Y_{-2}^{(3)}}   ^  {Y_{-1}^{(3)}}   ^  {Y_{0}^{(3)}}   ^  {Y_{1}^{(3)}}   ^  {Y_{2}^{(3)}}   ^  {Y_{3}^{(3)}}   ^ 
 +^ f_{\text{xyz}} | 0 | \frac{i}{\sqrt{2}} | 0 | 0 | 0 | -\frac{i}{\sqrt{2}} | 0 | 
 +^ f_{x\left(5x^2-r^2\right)} | \frac{\sqrt{5}}{4} | 0 | -\frac{\sqrt{3}}{4} | 0 | \frac{\sqrt{3}}{4} | 0 | -\frac{\sqrt{5}}{4} | 
 +^ f_{y\left(5y^2-r^2\right)} | -\frac{i \sqrt{5}}{4} | 0 | -\frac{i \sqrt{3}}{4} | 0 | -\frac{i \sqrt{3}}{4} | 0 | -\frac{i \sqrt{5}}{4} | 
 +^ f_{z\left(5z^2-r^2\right)} | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 
 +^ f_{x\left(y^2-z^2\right)} | -\frac{\sqrt{3}}{4} | 0 | -\frac{\sqrt{5}}{4} | 0 | \frac{\sqrt{5}}{4} | 0 | \frac{\sqrt{3}}{4} | 
 +^ f_{y\left(z^2-x^2\right)} | -\frac{i \sqrt{3}}{4} | 0 | \frac{i \sqrt{5}}{4} | 0 | \frac{i \sqrt{5}}{4} | 0 | -\frac{i \sqrt{3}}{4} | 
 +^ f_{z\left(x^2-y^2\right)} | 0 | \frac{1}{\sqrt{2}} | 0 | 0 | 0 | \frac{1}{\sqrt{2}} | 0 | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Irriducible representations and their onsite energy** > 
 + 
 +### 
 + 
 +^ ^\text{Eappxyz} | {{:physics_chemistry:pointgroup:cs_z_orb_3_1.png?150}} | 
 +|\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{Eappx3} | {{:physics_chemistry:pointgroup:cs_z_orb_3_2.png?150}} | 
 +|\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{Eappy3} | {{:physics_chemistry:pointgroup:cs_z_orb_3_3.png?150}} | 
 +|\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{Eappz3} | {{:physics_chemistry:pointgroup:cs_z_orb_3_4.png?150}} | 
 +|\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{Eappxy2z2} | {{:physics_chemistry:pointgroup:cs_z_orb_3_5.png?150}} | 
 +|\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}} |-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right) | ::: | 
 +|\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}} |-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right) | ::: | 
 +^ ^\text{Eappyz2x2} | {{:physics_chemistry:pointgroup:cs_z_orb_3_6.png?150}} | 
 +|\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{Eappzx2y2} | {{:physics_chemistry:pointgroup:cs_z_orb_3_7.png?150}} | 
 +|\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) | ::: | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +===== Coupling between two shells ===== 
 + 
 + 
 + 
 +### 
 + 
 +Click on one of the subsections to expand it or <hiddenSwitch expand all>  
 + 
 +### 
 + 
 +==== Potential for s-p orbital mixing ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + 0 & k\neq 1\lor (m\neq -1\land m\neq 1) \\ 
 + -A(1,1)+i B(1,1) & k=1\land m=-1 \\ 
 + A(1,1)+i B(1,1) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{0, k != 1 || (m != -1 && m != 1)}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}}, A[1, 1] + I*B[1, 1]] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,  
 +       {1, 1, A(1,1) + (I)*(B(1,1))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** > 
 + 
 +### 
 + 
 +|      ^  {Y_{-1}^{(1)}}   ^  {Y_{0}^{(1)}}   ^  {Y_{1}^{(1)}}   ^ 
 +^ {Y_{0}^{(0)}} | -\frac{A(1,1)+i B(1,1)}{\sqrt{3}} | 0 | \frac{A(1,1)-i B(1,1)}{\sqrt{3}} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  p_x   ^  p_y   ^  p_z   ^ 
 +^ \text{s} | -\sqrt{\frac{2}{3}} A(1,1) | \sqrt{\frac{2}{3}} B(1,1) | 0 | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for s-d orbital mixing ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ 
 + A(2,2)-i B(2,2) & k=2\land m=-2 \\ 
 + A(2,0) & k=2\land m=0 \\ 
 + A(2,2)+i B(2,2) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != 0 && m != 2)}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}}, A[2, 2] + I*B[2, 2]] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{2, 0, A(2,0)} ,  
 +       {2,-2, A(2,2) + (-I)*(B(2,2))} ,  
 +       {2, 2, A(2,2) + (I)*(B(2,2))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **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)+i B(2,2)}{\sqrt{5}} | 0 | \frac{A(2,0)}{\sqrt{5}} | 0 | \frac{A(2,2)-i B(2,2)}{\sqrt{5}} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  d_{x^2-y^2}   ^  d_{3z^2-r^2}   ^  d_{\text{yz}}   ^  d_{\text{xz}}   ^  d_{\text{xy}}   ^ 
 +^ \text{s} | \sqrt{\frac{2}{5}} A(2,2) | \frac{A(2,0)}{\sqrt{5}} | 0 | 0 | -\sqrt{\frac{2}{5}} B(2,2) | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for s-f orbital mixing ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + 0 & k\neq 3\lor (m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3) \\ 
 + -A(3,3)+i B(3,3) & k=3\land m=-3 \\ 
 + -A(3,1)+i B(3,1) & k=3\land m=-1 \\ 
 + A(3,1)+i B(3,1) & k=3\land m=1 \\ 
 + A(3,3)+i B(3,3) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != -1 && m != 1 && m != 3)}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}}, A[3, 3] + I*B[3, 3]] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,  
 +       {3, 1, A(3,1) + (I)*(B(3,1))} ,  
 +       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,  
 +       {3, 3, A(3,3) + (I)*(B(3,3))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **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{A(3,3)+i B(3,3)}{\sqrt{7}} | 0 | -\frac{A(3,1)+i B(3,1)}{\sqrt{7}} | 0 | \frac{A(3,1)-i B(3,1)}{\sqrt{7}} | 0 | \frac{A(3,3)-i B(3,3)}{\sqrt{7}} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  f_{\text{xyz}}   ^  f_{x\left(5x^2-r^2\right)}   ^  f_{y\left(5y^2-r^2\right)}   ^  f_{z\left(5z^2-r^2\right)}   ^  f_{x\left(y^2-z^2\right)}   ^  f_{y\left(z^2-x^2\right)}   ^  f_{z\left(x^2-y^2\right)}   ^ 
 +^ \text{s} | 0 | \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) | -\frac{\sqrt{3} B(3,1)+\sqrt{5} B(3,3)}{2 \sqrt{7}} | 0 | \frac{\sqrt{5} A(3,1)+\sqrt{3} A(3,3)}{2 \sqrt{7}} | \frac{1}{14} \left(\sqrt{35} B(3,1)-\sqrt{21} B(3,3)\right) | 0 | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for p-d orbital mixing ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + 0 & (k\neq 3\land (k\neq 1\lor (m\neq -1\land m\neq 1)))\lor (m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3) \\ 
 + -A(1,1)+i B(1,1) & k=1\land m=-1 \\ 
 + A(1,1)+i B(1,1) & k=1\land m=1 \\ 
 + -A(3,3)+i B(3,3) & k=3\land m=-3 \\ 
 + -A(3,1)+i B(3,1) & k=3\land m=-1 \\ 
 + A(3,1)+i B(3,1) & k=3\land m=1 \\ 
 + A(3,3)+i B(3,3) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || (m != -1 && m != 1))) || (m != -3 && m != -1 && m != 1 && m != 3)}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}}, A[3, 3] + I*B[3, 3]] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,  
 +       {1, 1, A(1,1) + (I)*(B(1,1))} ,  
 +       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,  
 +       {3, 1, A(3,1) + (I)*(B(3,1))} ,  
 +       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,  
 +       {3, 3, A(3,3) + (I)*(B(3,3))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **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}{7} \sqrt{\frac{3}{5}} (A(3,1)+i B(3,1))-\sqrt{\frac{2}{5}} (A(1,1)+i B(1,1)) | 0 | \frac{1}{105} \left(7 \sqrt{15} (A(1,1)-i B(1,1))-9 \sqrt{10} (A(3,1)-i B(3,1))\right) | 0 | -\frac{3}{7} (A(3,3)-i B(3,3)) | 
 +^ {Y_{0}^{(1)}} | 0 | -\frac{A(1,1)+i B(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (A(3,1)+i B(3,1)) | 0 | \frac{7 A(1,1)+2 \sqrt{6} A(3,1)-i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} | 0 | 
 +^ {Y_{1}^{(1)}} | \frac{3}{7} (A(3,3)+i B(3,3)) | 0 | \frac{3}{7} \sqrt{\frac{2}{5}} (A(3,1)+i B(3,1))-\frac{A(1,1)+i B(1,1)}{\sqrt{15}} | 0 | \sqrt{\frac{2}{5}} (A(1,1)-i B(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)-i B(3,1)) | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  d_{x^2-y^2}   ^  d_{3z^2-r^2}   ^  d_{\text{yz}}   ^  d_{\text{xz}}   ^  d_{\text{xy}}   ^ 
 +^ p_x | \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)-15 A(3,3)\right) | \sqrt{\frac{2}{15}} A(1,1)-\frac{6 A(3,1)}{7 \sqrt{5}} | 0 | 0 | \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} B(3,1)+\frac{3}{7} B(3,3) | 
 +^ p_y | \frac{1}{35} \left(-7 \sqrt{10} B(1,1)+\sqrt{15} B(3,1)+15 B(3,3)\right) | \frac{6 B(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} B(1,1) | 0 | 0 | \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) | 
 +^ p_z | 0 | 0 | \sqrt{\frac{2}{5}} B(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} B(3,1) | -\frac{1}{7} \sqrt{\frac{2}{5}} \left(7 A(1,1)+2 \sqrt{6} A(3,1)\right) | 0 | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for p-f orbital mixing ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + 0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ 
 + A(2,2)-i B(2,2) & k=2\land m=-2 \\ 
 + A(2,0) & k=2\land m=0 \\ 
 + A(2,2)+i B(2,2) & k=2\land m=2 \\ 
 + A(4,4)-i B(4,4) & k=4\land m=-4 \\ 
 + A(4,2)-i B(4,2) & k=4\land m=-2 \\ 
 + A(4,0) & k=4\land m=0 \\ 
 + A(4,2)+i B(4,2) & k=4\land m=2 \\ 
 + A(4,4)+i B(4,4) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}}, A[4, 4] + I*B[4, 4]] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{2, 0, A(2,0)} ,  
 +       {2,-2, A(2,2) + (-I)*(B(2,2))} ,  
 +       {2, 2, A(2,2) + (I)*(B(2,2))} ,  
 +       {4, 0, A(4,0)} ,  
 +       {4,-2, A(4,2) + (-I)*(B(4,2))} ,  
 +       {4, 2, A(4,2) + (I)*(B(4,2))} ,  
 +       {4,-4, A(4,4) + (-I)*(B(4,4))} ,  
 +       {4, 4, A(4,4) + (I)*(B(4,4))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **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)+i B(2,2))}{\sqrt{35}}-\frac{A(4,2)+i B(4,2)}{3 \sqrt{21}} | 0 | \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i B(4,2)) | 0 | -\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}} | 
 +^ {Y_{0}^{(1)}} | 0 | \sqrt{\frac{3}{35}} (A(2,2)+i B(2,2))+\frac{2 (A(4,2)+i B(4,2))}{3 \sqrt{7}} | 0 | \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} | 0 | \sqrt{\frac{3}{35}} (A(2,2)-i B(2,2))+\frac{2 (A(4,2)-i B(4,2))}{3 \sqrt{7}} | 0 | 
 +^ {Y_{1}^{(1)}} | -\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}} | 0 | \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i B(4,2)) | 0 | \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) | 0 | \frac{3 (A(2,2)-i B(2,2))}{\sqrt{35}}-\frac{A(4,2)-i B(4,2)}{3 \sqrt{21}} | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  f_{\text{xyz}}   ^  f_{x\left(5x^2-r^2\right)}   ^  f_{y\left(5y^2-r^2\right)}   ^  f_{z\left(5z^2-r^2\right)}   ^  f_{x\left(y^2-z^2\right)}   ^  f_{y\left(z^2-x^2\right)}   ^  f_{z\left(x^2-y^2\right)}   ^ 
 +^ p_x | 0 | \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) | \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)+7 B(4,4)\right)\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) | \sqrt{\frac{6}{35}} B(2,2)-\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} | 0 | 
 +^ p_y | 0 | \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)-7 B(4,4)\right)\right) | \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 | -\sqrt{\frac{6}{35}} B(2,2)+\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} | \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) | 0 | 
 +^ p_z | -\sqrt{\frac{6}{35}} B(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} B(4,2) | 0 | 0 | \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} | 0 | 0 | \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +==== Potential for d-f orbital mixing ==== 
 + 
 +<hidden **Potential parameterized with onsite energies of irriducible representations** > 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + 0 & (k\neq 5\land (((k\neq 1\lor (m\neq -1\land m\neq 1))\land k\neq 3)\lor (m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3)))\lor (m\neq -5\land m\neq -3\land m\neq -1\land m\neq 1\land m\neq 3\land m\neq 5) \\ 
 + -A(1,1)+i B(1,1) & k=1\land m=-1 \\ 
 + A(1,1)+i B(1,1) & k=1\land m=1 \\ 
 + -A(3,3)+i B(3,3) & k=3\land m=-3 \\ 
 + -A(3,1)+i B(3,1) & k=3\land m=-1 \\ 
 + A(3,1)+i B(3,1) & k=3\land m=1 \\ 
 + A(3,3)+i B(3,3) & k=3\land m=3 \\ 
 + -A(5,5)+i B(5,5) & k=5\land m=-5 \\ 
 + -A(5,3)+i B(5,3) & k=5\land m=-3 \\ 
 + -A(5,1)+i B(5,1) & k=5\land m=-1 \\ 
 + A(5,1)+i B(5,1) & k=5\land m=1 \\ 
 + A(5,3)+i B(5,3) & k=5\land m=3 \\ 
 + A(5,5)+i B(5,5) & \text{True} 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{0, (k != 5 && (((k != 1 || (m != -1 && m != 1)) && k != 3) || (m != -3 && m != -1 && m != 1 && m != 3))) || (m != -5 && m != -3 && m != -1 && m != 1 && m != 3 && m != 5)}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}}, A[5, 5] + I*B[5, 5]] 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden><hidden **Input format suitable for Quanty** > 
 + 
 +### 
 + 
 +<code Quanty Akm_Cs_Z.Quanty> 
 + 
 +Akm = {{1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,  
 +       {1, 1, A(1,1) + (I)*(B(1,1))} ,  
 +       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,  
 +       {3, 1, A(3,1) + (I)*(B(3,1))} ,  
 +       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,  
 +       {3, 3, A(3,3) + (I)*(B(3,3))} ,  
 +       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,  
 +       {5, 1, A(5,1) + (I)*(B(5,1))} ,  
 +       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,  
 +       {5, 3, A(5,3) + (I)*(B(5,3))} ,  
 +       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,  
 +       {5, 5, A(5,5) + (I)*(B(5,5))} } 
 + 
 +</code> 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **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)}} | -\sqrt{\frac{3}{7}} (A(1,1)+i B(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)+i B(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)+i B(5,1)) | 0 | \frac{33 \sqrt{35} (A(1,1)-i B(1,1))-22 \sqrt{210} (A(3,1)-i B(3,1))+25 \sqrt{21} (A(5,1)-i B(5,1))}{1155} | 0 | \frac{5}{33} \sqrt{2} (A(5,3)-i B(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)-i B(3,3)) | 0 | \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i B(5,5)) | 
 +^ {Y_{-1}^{(2)}} | 0 | -\sqrt{\frac{2}{7}} (A(1,1)+i B(1,1))-\frac{A(3,1)+i B(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (A(5,1)+i B(5,1)) | 0 | \frac{33 \sqrt{105} (A(1,1)-i B(1,1))-11 \sqrt{70} (A(3,1)-i B(3,1))-100 \sqrt{7} (A(5,1)-i B(5,1))}{1155} | 0 | -\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i B(3,3))-\frac{4}{33} \sqrt{5} (A(5,3)-i B(5,3)) | 0 | 
 +^ {Y_{0}^{(2)}} | \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i B(3,3))-\frac{2}{33} \sqrt{5} (A(5,3)+i B(5,3)) | 0 | -\sqrt{\frac{6}{35}} (A(1,1)+i B(1,1))-\frac{A(3,1)+i B(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (A(5,1)+i B(5,1)) | 0 | \frac{1}{385} \left(11 \sqrt{210} (A(1,1)-i B(1,1))+11 \sqrt{35} (A(3,1)-i B(3,1))+25 \sqrt{14} (A(5,1)-i B(5,1))\right) | 0 | \frac{2}{33} \sqrt{5} (A(5,3)-i B(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i B(3,3)) | 
 +^ {Y_{1}^{(2)}} | 0 | \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i B(3,3))+\frac{4}{33} \sqrt{5} (A(5,3)+i B(5,3)) | 0 | -\sqrt{\frac{3}{35}} (A(1,1)+i B(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (A(3,1)+i B(3,1))+\frac{20 (A(5,1)+i B(5,1))}{33 \sqrt{7}} | 0 | \frac{1}{231} \left(33 \sqrt{14} (A(1,1)-i B(1,1))+11 \sqrt{21} (A(3,1)-i B(3,1))-2 \sqrt{210} (A(5,1)-i B(5,1))\right) | 0 | 
 +^ {Y_{2}^{(2)}} | -\frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)+i B(5,5)) | 0 | \frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)+i B(3,3))-\frac{5}{33} \sqrt{2} (A(5,3)+i B(5,3)) | 0 | -\frac{A(1,1)+i B(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (A(3,1)+i B(3,1))-\frac{5 (A(5,1)+i B(5,1))}{11 \sqrt{21}} | 0 | \sqrt{\frac{3}{7}} (A(1,1)-i B(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)-i B(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)-i B(5,1)) | 
 + 
 + 
 +### 
 + 
 +</hidden> 
 +<hidden **The Hamiltonian on a basis of symmetric functions** > 
 + 
 +### 
 + 
 +|      ^  f_{\text{xyz}}   ^  f_{x\left(5x^2-r^2\right)}   ^  f_{y\left(5y^2-r^2\right)}   ^  f_{z\left(5z^2-r^2\right)}   ^  f_{x\left(y^2-z^2\right)}   ^  f_{y\left(z^2-x^2\right)}   ^  f_{z\left(x^2-y^2\right)}   ^ 
 +^ d_{x^2-y^2} | 0 | \frac{-99 \sqrt{210} A(1,1)+121 \sqrt{35} A(3,1)-5 \left(11 \sqrt{21} A(3,3)+10 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)+35 \sqrt{15} A(5,5)\right)}{2310} | \frac{-99 \sqrt{210} B(1,1)+121 \sqrt{35} B(3,1)+55 \sqrt{21} B(3,3)-50 \sqrt{14} B(5,1)-175 \sqrt{3} B(5,3)-175 \sqrt{15} B(5,5)}{2310} | 0 | \frac{1}{462} \left(33 \sqrt{14} A(1,1)+11 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)-2 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)+105 A(5,5)\right) | \frac{1}{462} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)-11 \sqrt{35} B(3,3)+2 \sqrt{210} B(5,1)+35 \sqrt{5} B(5,3)-105 B(5,5)\right) | 0 | 
 +^ d_{3z^2-r^2} | 0 | \frac{99 \sqrt{70} A(1,1)+33 \sqrt{105} A(3,1)+275 \sqrt{7} A(3,3)+75 \sqrt{42} A(5,1)-350 A(5,3)}{2310} | \frac{-99 \sqrt{70} B(1,1)-33 \sqrt{105} B(3,1)+275 \sqrt{7} B(3,3)-75 \sqrt{42} B(5,1)-350 B(5,3)}{2310} | 0 | \frac{1}{462} \left(33 \sqrt{42} A(1,1)+33 \sqrt{7} A(3,1)-11 \sqrt{105} A(3,3)+15 \sqrt{70} A(5,1)+14 \sqrt{15} A(5,3)\right) | \frac{1}{462} \left(33 \sqrt{42} B(1,1)+33 \sqrt{7} B(3,1)+11 \sqrt{105} B(3,3)+15 \sqrt{70} B(5,1)-14 \sqrt{15} B(5,3)\right) | 0 | 
 +^ d_{\text{yz}} | \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)\right) | 0 | 0 | -\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 | 0 | \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) | 
 +^ 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 | \sqrt{\frac{6}{35}} A(1,1)-\frac{2 A(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} A(5,1) | 0 | 0 | \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(-11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)-28 A(5,3)\right)\right) | 
 +^ d_{\text{xy}} | 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} | \frac{66 \sqrt{210} A(1,1)+11 \sqrt{35} A(3,1)+5 \left(11 \sqrt{21} A(3,3)-5 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)-35 \sqrt{15} A(5,5)\right)}{2310} | 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) | \frac{1}{462} \left(66 \sqrt{14} A(1,1)-33 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)+3 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)-105 A(5,5)\right) | 0 | 
 + 
 + 
 +###
  
 +</hidden>
  
 ===== Table of several point groups ===== ===== Table of several point groups =====

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