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| physics_chemistry:point_groups:cs:orientation_z [2018/03/24 00:03] – Maurits W. Haverkort | physics_chemistry:point_groups:cs:orientation_z [2018/04/06 09:16] (current) – Maurits W. Haverkort |
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| | ~~CLOSETOC~~ |
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| ====== 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 $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics $C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$. 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 $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ |
| | The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Cs Point group with orientation Z the form of the expansion coefficients is: |
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| ### | ### |
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| ==== 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}$$ | \end{cases}$$ |
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| | ### |
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| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
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| | ### |
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| | <code Quanty Akm_Cs_Z.Quanty.nb> |
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| | 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] |
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| | </code> |
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| ### | ### |
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| ### | ### |
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| | <code Quanty Akm_Cs_Z.Quanty> |
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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> |
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| ### | ### |
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| ### | ### |
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| | The operator representing the potential in second quantisation is given as: |
| | $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| | For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. |
| | $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ |
| | Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$ |
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| | ### |
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| | ### |
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| | we can express the operator as |
| | $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
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| | ### |
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| | ### |
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| | The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter. |
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| | ### |
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| | ### |
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| | | $ $ ^ $ {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) $| |
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| ### | ### |
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| | 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 |
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| | ### |
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| | ### |
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| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| |
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| ### | ### |
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| | After rotation we find |
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| ### | ### |
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| ===== Potential for s orbitals ===== | |
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| ===== Potential for p orbitals ===== | |
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| ===== Potential for d orbitals ===== | ### |
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| ===== Potential for f orbitals ===== | | $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ 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} $|$ \text{Ass}(0,0) $|$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$|$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$|$\color{darkred}{ 0 }$|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$|$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$|$\color{darkred}{ 0 }$| |
| | ^$ p_x $|$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $|$ 0 $|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$|$\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ 0 $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_y $|$\color{darkred}{ \sqrt{\frac{2}{3}} \text{Bsp}(1,1) }$|$ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$|$\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $| |
| | ^$ d_{x^2-y^2} $|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\frac{3}{7} \text{Apd}(3,3) }$|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$|$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3z^2-r^2} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)-\frac{6 \text{Apd}(3,1)}{7 \sqrt{5}} }$|$\color{darkred}{ \frac{6 \text{Bpd}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Bpd}(1,1) }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$|$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$|$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1) }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $|$ 0 $|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$| |
| | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$|$ 0 $|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$| |
| | ^$ d_{\text{xy}} $|$ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Bpd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Bpd}(3,1)+\frac{3}{7} \text{Bpd}(3,3) }$|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)+\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{7} \text{Apd}(3,3) }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,3) }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ 0 $|$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)+\frac{11 \text{Adf}(3,1)}{6 \sqrt{35}}-\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1)+\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}+\frac{1}{2} \sqrt{\frac{3}{35}} \text{Adf}(3,1)+\frac{5 \text{Adf}(3,3)}{6 \sqrt{7}}+\frac{5}{11} \sqrt{\frac{3}{14}} \text{Adf}(5,1)-\frac{5}{33} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}+\frac{5 \text{Bdf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}+\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,3) }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$\color{darkred}{ -3 \sqrt{\frac{3}{70}} \text{Bdf}(1,1)+\frac{11 \text{Bdf}(3,1)}{6 \sqrt{35}}+\frac{\text{Bdf}(3,3)}{2 \sqrt{21}}-\frac{5}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1)-\frac{5 \text{Bdf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ -\frac{3 \text{Bdf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{3}{35}} \text{Bdf}(3,1)+\frac{5 \text{Bdf}(3,3)}{6 \sqrt{7}}-\frac{5}{11} \sqrt{\frac{3}{14}} \text{Bdf}(5,1)-\frac{5}{33} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{\text{Adf}(3,3)}{2 \sqrt{21}}-\frac{5 \text{Adf}(5,1)}{33 \sqrt{14}}-\frac{5 \text{Adf}(5,3)}{22 \sqrt{3}}-\frac{5}{22} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$|$ 0 $|$ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Bdf}(1,1)+\frac{2 \text{Bdf}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Bdf}(5,1) }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ 0 }$|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Asf}(3,1)+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Asf}(3,3) }$|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{14}}+\frac{\text{Adf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{1}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)+\frac{5}{22} \text{Adf}(5,5) }$|$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{2 \sqrt{7}}-\frac{1}{2} \sqrt{\frac{5}{21}} \text{Adf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Adf}(5,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bdf}(3,1)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Bdf}(5,1)-\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ \frac{1}{2} \sqrt{\frac{5}{7}} \text{Bsf}(3,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Bsf}(3,3) }$|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ -\frac{\text{Bdf}(1,1)}{\sqrt{14}}-\frac{\text{Bdf}(3,1)}{2 \sqrt{21}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{1}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{5}{66} \sqrt{5} \text{Bdf}(5,3)-\frac{5}{22} \text{Bdf}(5,5) }$|$\color{darkred}{ \sqrt{\frac{3}{14}} \text{Bdf}(1,1)+\frac{\text{Bdf}(3,1)}{2 \sqrt{7}}+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Bdf}(3,3)+\frac{5}{11} \sqrt{\frac{5}{14}} \text{Bdf}(5,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,1)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{1}{11} \sqrt{\frac{15}{14}} \text{Adf}(5,1)+\frac{5}{66} \sqrt{5} \text{Adf}(5,3)-\frac{5}{22} \text{Adf}(5,5) }$|$ 0 $|$ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ 0 $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bdf}(1,1)-\frac{\text{Bdf}(3,1)}{\sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Bdf}(5,1)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $|$ 0 $|$ 0 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $| |
| |
| ===== 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 $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$. |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <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 ===== |
