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| — | physics_chemistry:point_groups:cs:orientation_y [2018/04/06 09:17] (current) – created Maurits W. Haverkort |
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| | ~~CLOSETOC~~ |
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| | ====== Orientation Y ====== |
| | |
| | ===== Symmetry Operations ===== |
| | |
| | ### |
| | |
| | In the Cs Point Group, with orientation Y there are the following symmetry operations |
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| | ### |
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| | ### |
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| | {{:physics_chemistry:pointgroup:cs_y.png}} |
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| | ### |
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| | ### |
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| | ^ Operator ^ Orientation ^ |
| | ^ $\text{E}$ | $\{0,0,0\}$ , | |
| | ^ $\sigma _h$ | $\{0,1,0\}$ , | |
| | |
| | ### |
| | |
| | ===== Different Settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:cs:orientation_x|Point Group Cs with orientation X]] |
| | * [[physics_chemistry:point_groups:cs:orientation_y|Point Group Cs with orientation Y]] |
| | * [[physics_chemistry:point_groups:cs:orientation_z|Point Group Cs with orientation Z]] |
| | |
| | ### |
| | |
| | ===== Character Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ \sigma_h \,{\text{(1)}} $ ^ |
| | ^ $ \text{A'} $ | $ 1 $ | $ 1 $ | |
| | ^ $ \text{A''} $ | $ 1 $ | $ -1 $ | |
| | |
| | ### |
| | |
| | ===== Product Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{A'} $ ^ $ \text{A''} $ ^ |
| | ^ $ \text{A'} $ | $ \text{A'} $ | $ \text{A''} $ | |
| | ^ $ \text{A''} $ | $ \text{A''} $ | $ \text{A'} $ | |
| | |
| | ### |
| | |
| | ===== Sub Groups with compatible settings ===== |
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| | ### |
| | |
| | * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] |
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| | ### |
| | |
| | ===== Super Groups with compatible settings ===== |
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| | ### |
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| | * [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:c3v:orientation_zy|Point Group C3v with orientation Zy]] |
| | * [[physics_chemistry:point_groups:c4v:orientation_zxy|Point Group C4v with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:c6v:orientation_zx|Point Group C6v with orientation Zx]] |
| | * [[physics_chemistry:point_groups:c6v:orientation_zy|Point Group C6v with orientation Zy]] |
| | * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_zy|Point Group D3d with orientation Zy]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_zy_a|Point Group D3d with orientation Zy_A]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_zy_b|Point Group D3d with orientation Zy_B]] |
| | * [[physics_chemistry:point_groups:d3h:orientation_zx|Point Group D3h with orientation Zx]] |
| | * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:d5d:orientation_zy|Point Group D5d with orientation Zy]] |
| | * [[physics_chemistry:point_groups:d5h:orientation_zx|Point Group D5h 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:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]] |
| | * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]] |
| | * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:th:orientation_xyz|Point Group Th with orientation xyz]] |
| | |
| | ### |
| | |
| | ===== Invariant Potential expanded on renormalized spherical Harmonics ===== |
| | |
| | ### |
| | |
| | Any potential (function) can be written as a sum over spherical harmonics. |
| | $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ |
| | Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ |
| | The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Cs Point group with orientation Y the form of the expansion coefficients is: |
| | |
| | ### |
| | |
| | ==== Expansion ==== |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1) & k=1\land m=-1 \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(1,1) & k=1\land m=1 \\ |
| | A(2,2) & k=2\land (m=-2\lor m=2) \\ |
| | -A(2,1) & k=2\land m=-1 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,1) & k=2\land m=1 \\ |
| | -A(3,3) & k=3\land m=-3 \\ |
| | A(3,2) & k=3\land (m=-2\lor m=2) \\ |
| | -A(3,1) & k=3\land m=-1 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,1) & k=3\land m=1 \\ |
| | A(3,3) & k=3\land m=3 \\ |
| | A(4,4) & k=4\land (m=-4\lor m=4) \\ |
| | -A(4,3) & k=4\land m=-3 \\ |
| | A(4,2) & k=4\land (m=-2\lor m=2) \\ |
| | -A(4,1) & k=4\land m=-1 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,1) & k=4\land m=1 \\ |
| | A(4,3) & k=4\land m=3 \\ |
| | -A(5,5) & k=5\land m=-5 \\ |
| | A(5,4) & k=5\land (m=-4\lor m=4) \\ |
| | -A(5,3) & k=5\land m=-3 \\ |
| | A(5,2) & k=5\land (m=-2\lor m=2) \\ |
| | -A(5,1) & k=5\land m=-1 \\ |
| | A(5,0) & k=5\land m=0 \\ |
| | A(5,1) & k=5\land m=1 \\ |
| | A(5,3) & k=5\land m=3 \\ |
| | A(5,5) & k=5\land m=5 \\ |
| | A(6,6) & k=6\land (m=-6\lor m=6) \\ |
| | -A(6,5) & k=6\land m=-5 \\ |
| | A(6,4) & k=6\land (m=-4\lor m=4) \\ |
| | -A(6,3) & k=6\land m=-3 \\ |
| | A(6,2) & k=6\land (m=-2\lor m=2) \\ |
| | -A(6,1) & k=6\land m=-1 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,1) & k=6\land m=1 \\ |
| | A(6,3) & k=6\land m=3 \\ |
| | A(6,5) & k=6\land m=5 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1], k == 1 && m == 1}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {-A[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1], k == 2 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}, {A[3, 3], k == 3 && m == 3}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {-A[4, 3], k == 4 && m == -3}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {-A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1], k == 4 && m == 1}, {A[4, 3], k == 4 && m == 3}, {-A[5, 5], k == 5 && m == -5}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {-A[5, 3], k == 5 && m == -3}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {-A[5, 1], k == 5 && m == -1}, {A[5, 0], k == 5 && m == 0}, {A[5, 1], k == 5 && m == 1}, {A[5, 3], k == 5 && m == 3}, {A[5, 5], k == 5 && m == 5}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {-A[6, 5], k == 6 && m == -5}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {-A[6, 3], k == 6 && m == -3}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {-A[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {A[6, 1], k == 6 && m == 1}, {A[6, 3], k == 6 && m == 3}, {A[6, 5], k == 6 && m == 5}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | ==== Input format suitable for Quanty ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = { 0, A(0,0)} , |
| | {1, 0, A(1,0)} , |
| | {1,-1, (-1)*(A(1,1))} , |
| | {1, 1, A(1,1)} , |
| | {2, 0, A(2,0)} , |
| | {2,-1, (-1)*(A(2,1))} , |
| | {2, 1, A(2,1)} , |
| | {2,-2, A(2,2)} , |
| | {2, 2, A(2,2)} , |
| | {3, 0, A(3,0)} , |
| | {3,-1, (-1)*(A(3,1))} , |
| | {3, 1, A(3,1)} , |
| | {3,-2, A(3,2)} , |
| | {3, 2, A(3,2)} , |
| | {3,-3, (-1)*(A(3,3))} , |
| | {3, 3, A(3,3)} , |
| | {4, 0, A(4,0)} , |
| | {4,-1, (-1)*(A(4,1))} , |
| | {4, 1, A(4,1)} , |
| | {4,-2, A(4,2)} , |
| | {4, 2, A(4,2)} , |
| | {4,-3, (-1)*(A(4,3))} , |
| | {4, 3, A(4,3)} , |
| | {4,-4, A(4,4)} , |
| | {4, 4, A(4,4)} , |
| | {5, 0, A(5,0)} , |
| | {5,-1, (-1)*(A(5,1))} , |
| | {5, 1, A(5,1)} , |
| | {5,-2, A(5,2)} , |
| | {5, 2, A(5,2)} , |
| | {5,-3, (-1)*(A(5,3))} , |
| | {5, 3, A(5,3)} , |
| | {5,-4, A(5,4)} , |
| | {5, 4, A(5,4)} , |
| | {5,-5, (-1)*(A(5,5))} , |
| | {5, 5, A(5,5)} , |
| | {6, 0, A(6,0)} , |
| | {6,-1, (-1)*(A(6,1))} , |
| | {6, 1, A(6,1)} , |
| | {6,-2, A(6,2)} , |
| | {6, 2, A(6,2)} , |
| | {6,-3, (-1)*(A(6,3))} , |
| | {6, 3, A(6,3)} , |
| | {6,-4, A(6,4)} , |
| | {6, 4, A(6,4)} , |
| | {6,-5, (-1)*(A(6,5))} , |
| | {6, 5, A(6,5)} , |
| | {6,-6, A(6,6)} , |
| | {6, 6, A(6,6)} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | ==== One particle coupling on a basis of spherical harmonics ==== |
| | |
| | ### |
| | |
| | The operator representing the potential in second quantisation is given as: |
| | $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| | For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. |
| | $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ |
| | Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$ |
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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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| | ### |
| | |
| | | $ $ ^ $ {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)}{\sqrt{3}} }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ \frac{\text{Asp}(1,1)}{\sqrt{3}} }$|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$ -\frac{\text{Asd}(2,1)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,1)}{\sqrt{5}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ -\frac{\text{Asf}(3,3)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ -\frac{\text{Asf}(3,1)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,1)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ \frac{\text{Asf}(3,3)}{\sqrt{7}} }$| |
| | ^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ -\frac{\text{Asp}(1,1)}{\sqrt{3}} }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ -\frac{1}{5} \sqrt{3} \text{App}(2,1) $|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\sqrt{\frac{2}{5}} \text{Apd}(1,1) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{3}{7} \text{Apd}(3,3) }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ \frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} \text{Apf}(2,1) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{3 \text{Apf}(2,1)}{5 \sqrt{7}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,1) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ -\frac{1}{3} \text{Apf}(4,3) $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ -\frac{1}{5} \sqrt{3} \text{App}(2,1) $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ \frac{1}{5} \sqrt{3} \text{App}(2,1) $|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{\text{Apd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$ -\frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ -\frac{2}{5} \sqrt{\frac{6}{7}} \text{Apf}(2,1)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \frac{2}{5} \sqrt{\frac{6}{7}} \text{Apf}(2,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ \frac{\text{Apf}(4,3)}{3 \sqrt{3}} $| |
| | ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ \frac{\text{Asp}(1,1)}{\sqrt{3}} }$|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ \frac{1}{5} \sqrt{3} \text{App}(2,1) $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ \frac{3}{7} \text{Apd}(3,3) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1)-\frac{\text{Apd}(1,1)}{\sqrt{15}} }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ \frac{1}{3} \text{Apf}(4,3) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ \frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,1)-\frac{3 \text{Apf}(2,1)}{5 \sqrt{7}} $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,1)-\frac{\text{Apf}(4,1)}{3 \sqrt{7}} $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $| |
| | ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ \frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1)-\sqrt{\frac{2}{5}} \text{Apd}(1,1) }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{3}{7} \text{Apd}(3,3) }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ \frac{1}{21} \sqrt{5} \text{Add}(4,1)-\frac{1}{7} \sqrt{6} \text{Add}(2,1) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ -\sqrt{\frac{3}{7}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)-\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)+\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$| |
| | ^$ {Y_{-1}^{(2)}} $|$ -\frac{\text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ -\frac{\text{Apd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ \frac{1}{21} \sqrt{5} \text{Add}(4,1)-\frac{1}{7} \sqrt{6} \text{Add}(2,1) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ -\frac{1}{7} \text{Add}(2,1)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ \sqrt{\frac{3}{35}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)-\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ \frac{3}{7} \sqrt{\frac{2}{5}} \text{Apd}(3,1)-\frac{\text{Apd}(1,1)}{\sqrt{15}} }$|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ -\frac{1}{7} \text{Add}(2,1)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ \frac{1}{7} \text{Add}(2,1)+\frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{2}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \frac{2}{33} \sqrt{5} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3) }$| |
| | ^$ {Y_{1}^{(2)}} $|$ \frac{\text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{\text{Apd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ \frac{1}{7} \text{Add}(2,1)+\frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ \frac{1}{7} \sqrt{6} \text{Add}(2,1)-\frac{1}{21} \sqrt{5} \text{Add}(4,1) $|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)+\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ -\frac{3}{7} \text{Apd}(3,3) }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} \text{Apd}(3,2) }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ \frac{1}{7} \sqrt{6} \text{Add}(2,1)-\frac{1}{21} \sqrt{5} \text{Add}(4,1) $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ -\frac{\text{Adf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)-\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ \sqrt{\frac{3}{7}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)+\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$| |
| | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ -\frac{\text{Asf}(3,3)}{\sqrt{7}} }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ -\frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$\color{darkred}{ -\sqrt{\frac{3}{7}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)-\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{2}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ -\frac{1}{3} \text{Aff}(2,1)+\frac{1}{11} \sqrt{\frac{10}{3}} \text{Aff}(4,1)-\frac{5}{429} \sqrt{7} \text{Aff}(6,1) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ \frac{1}{11} \sqrt{7} \text{Aff}(4,3)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ -\frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $| |
| | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$ \frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} \text{Apf}(2,1) $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ \frac{1}{3} \text{Apf}(4,3) $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$ -\frac{1}{3} \text{Aff}(2,1)+\frac{1}{11} \sqrt{\frac{10}{3}} \text{Aff}(4,1)-\frac{5}{429} \sqrt{7} \text{Aff}(6,1) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ -\frac{\text{Aff}(2,1)}{\sqrt{15}}-\frac{4}{33} \sqrt{2} \text{Aff}(4,1)+\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,3)+\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $| |
| | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ -\frac{\text{Asf}(3,1)}{\sqrt{7}} }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ -\frac{2}{5} \sqrt{\frac{6}{7}} \text{Apf}(2,1)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$\color{darkred}{ \frac{\text{Adf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)+\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ -\sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ -\frac{\text{Aff}(2,1)}{\sqrt{15}}-\frac{4}{33} \sqrt{2} \text{Aff}(4,1)+\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ -\frac{1}{15} \sqrt{2} \text{Aff}(2,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,1)-\frac{25}{429} \sqrt{14} \text{Aff}(6,1) $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,3)-\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $| |
| | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ \frac{3 \text{Apf}(2,1)}{5 \sqrt{7}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,1) $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,1)-\frac{3 \text{Apf}(2,1)}{5 \sqrt{7}} $|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$\color{darkred}{ \sqrt{\frac{3}{35}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)-\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,1)+\frac{20 \text{Adf}(5,1)}{33 \sqrt{7}} }$|$\color{darkred}{ \frac{5}{33} \text{Adf}(5,2)-\frac{2 \text{Adf}(3,2)}{3 \sqrt{7}} }$|$ \frac{1}{11} \sqrt{7} \text{Aff}(4,3)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ -\frac{1}{15} \sqrt{2} \text{Aff}(2,1)-\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,1)-\frac{25}{429} \sqrt{14} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ \frac{1}{15} \sqrt{2} \text{Aff}(2,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,1)+\frac{25}{429} \sqrt{14} \text{Aff}(6,1) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ \frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3)-\frac{1}{11} \sqrt{7} \text{Aff}(4,3) $| |
| | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,1)}{\sqrt{7}} }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ \frac{2}{5} \sqrt{\frac{6}{7}} \text{Apf}(2,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3) }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)}{\sqrt{21}}-\frac{5 \text{Adf}(5,2)}{11 \sqrt{3}} }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ -\frac{\text{Adf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} \text{Adf}(3,1)-\frac{5 \text{Adf}(5,1)}{11 \sqrt{21}} }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,3)+\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $|$ \frac{1}{15} \sqrt{2} \text{Aff}(2,1)+\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,1)+\frac{25}{429} \sqrt{14} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ \frac{\text{Aff}(2,1)}{\sqrt{15}}+\frac{4}{33} \sqrt{2} \text{Aff}(4,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $| |
| | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)}{\sqrt{7}} }$|$ -\frac{1}{3} \text{Apf}(4,3) $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,1)-\frac{\text{Apf}(4,1)}{3 \sqrt{7}} $|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Adf}(1,1)+\frac{\text{Adf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} \text{Adf}(5,1) }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$ -\frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,3)-\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ \frac{\text{Aff}(2,1)}{\sqrt{15}}+\frac{4}{33} \sqrt{2} \text{Aff}(4,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \frac{1}{3} \text{Aff}(2,1)-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Aff}(4,1)+\frac{5}{429} \sqrt{7} \text{Aff}(6,1) $| |
| | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,3)}{\sqrt{7}} }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ \frac{\text{Apf}(4,3)}{3 \sqrt{3}} $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{2}{33} \sqrt{5} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3) }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,2)-\frac{1}{33} \sqrt{5} \text{Adf}(5,2) }$|$\color{darkred}{ \sqrt{\frac{3}{7}} \text{Adf}(1,1)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,1)+\frac{1}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,1) }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ \frac{10}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,3)-\frac{1}{11} \sqrt{7} \text{Aff}(4,3) $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ \frac{1}{3} \text{Aff}(2,1)-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Aff}(4,1)+\frac{5}{429} \sqrt{7} \text{Aff}(6,1) $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| |
| | |
| | |
| | ### |
| | |
| | ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== |
| | |
| | ### |
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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 |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | | $ $ ^ $ {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_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ 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 $| |
| | ^$ d_{z^2-x^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|$ \frac{\sqrt{3}}{2} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3y^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|$ -\frac{1}{2} $|$ 0 $|$ -\frac{\sqrt{\frac{3}{2}}}{2} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ 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 }$| |
| | ^$ 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_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| |
| | ^$ f_{y\left(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(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| |
| | ^$ 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} $| |
| | |
| | |
| | ### |
| | |
| | ==== One particle coupling on a basis of symmetry adapted functions ==== |
| | |
| | ### |
| | |
| | After rotation we find |
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| | ### |
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| | |
| | |
| | ### |
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| | | $ $ ^ $ \text{s} $ ^ $ p_z $ ^ $ p_x $ ^ $ p_y $ ^ $ d_{z^2-x^2} $ ^ $ d_{3y^2-r^2} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ f_{\text{xyz}} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ |
| | ^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$|$\color{darkred}{ 0 }$|$ \frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} $|$ -\frac{\text{Asd}(2,0)}{2 \sqrt{5}}-\sqrt{\frac{3}{10}} \text{Asd}(2,2) $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} \text{Asd}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\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}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$\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}{ 0 }$| |
| | ^$ p_z $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ -\frac{1}{5} \sqrt{6} \text{App}(2,1) $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}+\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)-\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) $|$ 0 $| |
| | ^$ p_x $|$\color{darkred}{ -\sqrt{\frac{2}{3}} \text{Asp}(1,1) }$|$ -\frac{1}{5} \sqrt{6} \text{App}(2,1) $|$ \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{Apd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{14} \text{Apd}(3,3) }$|$\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)+\frac{3 \text{Apd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Apd}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ -\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{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) $|$ -\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}} $|$ 0 $| |
| | ^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\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}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) $|$ 0 $|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $| |
| | ^$ d_{z^2-x^2} $|$ \frac{1}{2} \sqrt{\frac{3}{5}} \text{Asd}(2,0)-\frac{\text{Asd}(2,2)}{\sqrt{10}} $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}+\frac{9 \text{Apd}(3,0)}{14 \sqrt{5}}-\frac{1}{7} \sqrt{\frac{3}{2}} \text{Apd}(3,2) }$|$\color{darkred}{ \sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} \text{Apd}(3,1)+\frac{3}{14} \text{Apd}(3,3) }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{19}{84} \text{Add}(4,0)-\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{6} \sqrt{\frac{5}{14}} \text{Add}(4,4) $|$ -\frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)-\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) $|$ 0 $|$ 0 $|$ \frac{1}{6} \sqrt{\frac{5}{7}} \text{Add}(4,3)-\frac{1}{6} \sqrt{5} \text{Add}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)+\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)+\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,3)+\frac{5 \text{Adf}(5,1)}{6 \sqrt{14}}-\frac{5}{44} \sqrt{3} \text{Adf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{\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{17}{22} \sqrt{\frac{5}{42}} \text{Adf}(5,1)+\frac{1}{132} \sqrt{5} \text{Adf}(5,3)-\frac{5}{44} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3y^2-r^2} $|$ -\frac{\text{Asd}(2,0)}{2 \sqrt{5}}-\sqrt{\frac{3}{10}} \text{Asd}(2,2) $|$\color{darkred}{ -\frac{\text{Apd}(1,0)}{\sqrt{15}}-\frac{3}{14} \sqrt{\frac{3}{5}} \text{Apd}(3,0)-\frac{3 \text{Apd}(3,2)}{7 \sqrt{2}} }$|$\color{darkred}{ \sqrt{\frac{2}{15}} \text{Apd}(1,1)+\frac{3 \text{Apd}(3,1)}{14 \sqrt{5}}+\frac{3}{14} \sqrt{3} \text{Apd}(3,3) }$|$\color{darkred}{ 0 }$|$ -\frac{1}{7} \sqrt{3} \text{Add}(2,0)+\frac{1}{7} \sqrt{2} \text{Add}(2,2)-\frac{5 \text{Add}(4,0)}{28 \sqrt{3}}-\frac{1}{7} \sqrt{\frac{5}{6}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{42}} \text{Add}(4,4) $|$ \text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)+\frac{3}{28} \text{Add}(4,0)+\frac{1}{7} \sqrt{\frac{5}{2}} \text{Add}(4,2)+\frac{1}{2} \sqrt{\frac{5}{14}} \text{Add}(4,4) $|$ 0 $|$ 0 $|$ \frac{2}{7} \sqrt{2} \text{Add}(2,1)+\frac{1}{14} \sqrt{\frac{5}{3}} \text{Add}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Add}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}+\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }$|$\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{7}{15}} \text{Adf}(3,1)-\frac{\text{Adf}(3,3)}{6 \sqrt{7}}-\frac{5 \text{Adf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Adf}(5,3)+\frac{5}{44} \sqrt{5} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)+\frac{\text{Adf}(3,0)}{\sqrt{21}}-\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ -\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{3}{22} \sqrt{\frac{5}{14}} \text{Adf}(5,1)-\frac{7}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,3)-\frac{5}{44} \sqrt{3} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xy}} $|$ 0 $|$\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 $|$ 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{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{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}{ 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) }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ 0 $|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,1)+\frac{1}{21} \sqrt{5} \text{Add}(4,1)+\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \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{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{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$| |
| | ^$ d_{\text{xz}} $|$ -\sqrt{\frac{2}{5}} \text{Asd}(2,1) $|$\color{darkred}{ -\sqrt{\frac{2}{5}} \text{Apd}(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,1) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ \frac{1}{6} \sqrt{\frac{5}{7}} \text{Add}(4,3)-\frac{1}{6} \sqrt{5} \text{Add}(4,1) $|$ \frac{2}{7} \sqrt{2} \text{Add}(2,1)+\frac{1}{14} \sqrt{\frac{5}{3}} \text{Add}(4,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Add}(4,3) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{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}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\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}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}+\frac{1}{3} \text{Apf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ -\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 }$|$ \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{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ 0 $|$ 0 $|$ -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ 0 $|$\color{darkred}{ \frac{3}{2} \sqrt{\frac{3}{35}} \text{Adf}(1,0)+\frac{2 \text{Adf}(3,0)}{\sqrt{105}}+\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2)+\frac{5}{11} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ -\frac{3 \text{Adf}(1,0)}{2 \sqrt{35}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{35}}+\sqrt{\frac{2}{21}} \text{Adf}(3,2)-\frac{5}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0)-\frac{5 \text{Adf}(5,2)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{6}{35}} \text{Adf}(1,1)-\frac{2 \text{Adf}(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} \text{Adf}(5,1) }$|$ 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) $|$ \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ 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) $|$ \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) $|$ 0 $| |
| | ^$ 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}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{1}{2} \sqrt{\frac{5}{21}} \text{Apf}(4,1)-\frac{1}{6} \sqrt{\frac{5}{3}} \text{Apf}(4,3) $|$ -\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) $|$ 0 $|$\color{darkred}{ 3 \sqrt{\frac{3}{70}} \text{Adf}(1,1)-\frac{\text{Adf}(3,1)}{6 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(3,3)+\frac{5 \text{Adf}(5,1)}{6 \sqrt{14}}-\frac{5}{44} \sqrt{3} \text{Adf}(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ \frac{3 \text{Adf}(1,1)}{\sqrt{70}}-\frac{1}{2} \sqrt{\frac{7}{15}} \text{Adf}(3,1)-\frac{\text{Adf}(3,3)}{6 \sqrt{7}}-\frac{5 \text{Adf}(5,1)}{22 \sqrt{42}}-\frac{5}{132} \text{Adf}(5,3)+\frac{5}{44} \sqrt{5} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$ 0 $|$ \frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{1}{22} \sqrt{5} \text{Aff}(4,1)+\frac{1}{22} \sqrt{35} \text{Aff}(4,3)+\frac{25}{143} \sqrt{\frac{7}{6}} \text{Aff}(6,1)-\frac{5}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,3) $|$ \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{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ \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_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{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}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{\text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)-\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$ -\sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{\text{Apf}(4,1)}{3 \sqrt{7}}-\frac{1}{3} \text{Apf}(4,3) $|$ 0 $|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{2 \sqrt{7}}+\frac{\text{Adf}(3,0)}{3 \sqrt{7}}-\frac{5 \text{Adf}(5,0)}{66 \sqrt{7}}+\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ -\frac{1}{2} \sqrt{\frac{3}{7}} \text{Adf}(1,0)+\frac{\text{Adf}(3,0)}{\sqrt{21}}-\frac{5 \text{Adf}(5,0)}{22 \sqrt{21}}-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{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) }$|$ 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) $|$ -\frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{3}{22} \sqrt{3} \text{Aff}(4,1)+\frac{1}{22} \sqrt{\frac{7}{3}} \text{Aff}(4,3)-\frac{5}{429} \sqrt{70} \text{Aff}(6,1)+\frac{15}{286} \sqrt{7} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{35}{33}} \text{Aff}(6,5) $|$ 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) $|$ \frac{\text{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|$ 0 $| |
| | ^$ 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) }$|$ \sqrt{\frac{6}{35}} \text{Apf}(2,1)+\frac{5 \text{Apf}(4,1)}{6 \sqrt{7}}+\frac{1}{6} \text{Apf}(4,3) $|$ -\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}} $|$ 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{17}{22} \sqrt{\frac{5}{42}} \text{Adf}(5,1)+\frac{1}{132} \sqrt{5} \text{Adf}(5,3)-\frac{5}{44} \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{3}{22} \sqrt{\frac{5}{14}} \text{Adf}(5,1)-\frac{7}{44} \sqrt{\frac{5}{3}} \text{Adf}(5,3)-\frac{5}{44} \sqrt{3} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$ 0 $|$ \frac{\text{Aff}(2,1)}{3 \sqrt{10}}+\frac{5 \text{Aff}(4,1)}{22 \sqrt{3}}-\frac{1}{22} \sqrt{21} \text{Aff}(4,3)+\frac{25}{429} \sqrt{\frac{35}{2}} \text{Aff}(6,1)+\frac{5}{143} \sqrt{7} \text{Aff}(6,3) $|$ \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{Aff}(2,1)}{\sqrt{6}}+\frac{1}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)-\frac{5}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)+\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|$ \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_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \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}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$ -\frac{7}{66} \sqrt{5} \text{Aff}(4,1)+\frac{1}{66} \sqrt{35} \text{Aff}(4,3)+\frac{5}{143} \sqrt{\frac{21}{2}} \text{Aff}(6,1)+\frac{5}{286} \sqrt{105} \text{Aff}(6,3)-\frac{5}{26} \sqrt{\frac{7}{11}} \text{Aff}(6,5) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $| |
| | |
| | |
| | ### |
| | |
| | ===== Coupling for a single shell ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$. |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| | |
| | ==== Potential for s orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \text{Eap} & k=0\land m=0 \\ |
| | 0 & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{Eap, k == 0 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{0, 0, Eap} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \text{Eap} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{s} $ ^ |
| | ^$ \text{s} $|$ \text{Eap} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ \text{s} $|$ 1 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eap}$$ | {{:physics_chemistry:pointgroup:cs_y_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** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{3} (\text{Eapp}+\text{Eapx}+\text{Eapz}) & k=0\land m=0 \\ |
| | 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ |
| | -\frac{5 (\text{Eapp}-\text{Eapx})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\ |
| | \frac{5 \text{Mapzx}}{\sqrt{6}} & k=2\land m=-1 \\ |
| | -\frac{5}{6} (\text{Eapp}+\text{Eapx}-2 \text{Eapz}) & k=2\land m=0 \\ |
| | -\frac{5 \text{Mapzx}}{\sqrt{6}} & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eapp + Eapx + Eapz)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {(-5*(Eapp - Eapx))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}, {(5*Mapzx)/Sqrt[6], k == 2 && m == -1}, {(-5*(Eapp + Eapx - 2*Eapz))/6, k == 2 && m == 0}}, (-5*Mapzx)/Sqrt[6]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{0, 0, (1/3)*(Eapp + Eapx + Eapz)} , |
| | {2, 0, (-5/6)*(Eapp + Eapx + (-2)*(Eapz))} , |
| | {2, 1, (-5)*((1/(sqrt(6)))*(Mapzx))} , |
| | {2,-1, (5)*((1/(sqrt(6)))*(Mapzx))} , |
| | {2,-2, (-5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapx)))} , |
| | {2, 2, (-5/2)*((1/(sqrt(6)))*(Eapp + (-1)*(Eapx)))} } |
| | |
| | </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{Eapp}+\text{Eapx}}{2} $|$ \frac{\text{Mapzx}}{\sqrt{2}} $|$ \frac{\text{Eapp}-\text{Eapx}}{2} $| |
| | ^$ {Y_{0}^{(1)}} $|$ \frac{\text{Mapzx}}{\sqrt{2}} $|$ \text{Eapz} $|$ -\frac{\text{Mapzx}}{\sqrt{2}} $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{\text{Eapp}-\text{Eapx}}{2} $|$ -\frac{\text{Mapzx}}{\sqrt{2}} $|$ \frac{\text{Eapp}+\text{Eapx}}{2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_z $ ^ $ p_x $ ^ $ p_y $ ^ |
| | ^$ p_z $|$ \text{Eapz} $|$ \text{Mapzx} $|$ 0 $| |
| | ^$ p_x $|$ \text{Mapzx} $|$ \text{Eapx} $|$ 0 $| |
| | ^$ p_y $|$ 0 $|$ 0 $|$ \text{Eapp} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| |
| | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| |
| | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eapz}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_1_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | |
| | ^ ^$$\text{Eapx}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_1_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | |
| | ^ ^$$\text{Eapp}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_1_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{5} (\text{Eappxy}+\text{Eappyz}+\text{Eapxz}+\text{Eapy2}+\text{Eapz2x2}) & k=0\land m=0 \\ |
| | 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ |
| | \frac{1}{4} \left(-\sqrt{6} \text{Eappyz}+\sqrt{6} \text{Eapxz}-\sqrt{6} \text{Eapy2}+\sqrt{6} \text{Eapz2x2}+2 \sqrt{2} \text{Mapz2x2y2}\right) & k=2\land (m=-2\lor m=2) \\ |
| | \frac{\sqrt{3} \text{Mappxyyz}-2 \text{Mapy2xz}}{\sqrt{2}} & k=2\land m=-1 \\ |
| | \frac{1}{2} \left(-2 \text{Eappxy}+\text{Eappyz}+\text{Eapxz}-\text{Eapy2}+\text{Eapz2x2}-2 \sqrt{3} \text{Mapz2x2y2}\right) & k=2\land m=0 \\ |
| | \sqrt{2} \text{Mapy2xz}-\sqrt{\frac{3}{2}} \text{Mappxyyz} & k=2\land m=1 \\ |
| | -\frac{3}{8} \sqrt{\frac{7}{10}} \left(4 \text{Eappxy}-3 \text{Eapy2}-\text{Eapz2x2}-2 \sqrt{3} \text{Mapz2x2y2}\right) & k=4\land (m=-4\lor m=4) \\ |
| | -\frac{3}{4} \sqrt{\frac{7}{5}} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) & k=4\land m=-3 \\ |
| | -\frac{3 \left(4 \text{Eappyz}-4 \text{Eapxz}-3 \text{Eapy2}+3 \text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right)}{4 \sqrt{10}} & k=4\land (m=-2\lor m=2) \\ |
| | -\frac{3 \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}-7 \text{Mapz2x2xz}\right)}{4 \sqrt{5}} & k=4\land m=-1 \\ |
| | \frac{3}{40} \left(4 \text{Eappxy}-16 \text{Eappyz}-16 \text{Eapxz}+9 \text{Eapy2}+19 \text{Eapz2x2}-10 \sqrt{3} \text{Mapz2x2y2}\right) & k=4\land m=0 \\ |
| | \frac{3 \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}-7 \text{Mapz2x2xz}\right)}{4 \sqrt{5}} & k=4\land m=1 \\ |
| | \frac{3}{4} \sqrt{\frac{7}{5}} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eappxy + Eappyz + Eapxz + Eapy2 + Eapz2x2)/5, k == 0 && m == 0}, {0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(-(Sqrt[6]*Eappyz) + Sqrt[6]*Eapxz - Sqrt[6]*Eapy2 + Sqrt[6]*Eapz2x2 + 2*Sqrt[2]*Mapz2x2y2)/4, k == 2 && (m == -2 || m == 2)}, {(Sqrt[3]*Mappxyyz - 2*Mapy2xz)/Sqrt[2], k == 2 && m == -1}, {(-2*Eappxy + Eappyz + Eapxz - Eapy2 + Eapz2x2 - 2*Sqrt[3]*Mapz2x2y2)/2, k == 2 && m == 0}, {-(Sqrt[3/2]*Mappxyyz) + Sqrt[2]*Mapy2xz, k == 2 && m == 1}, {(-3*Sqrt[7/10]*(4*Eappxy - 3*Eapy2 - Eapz2x2 - 2*Sqrt[3]*Mapz2x2y2))/8, k == 4 && (m == -4 || m == 4)}, {(-3*Sqrt[7/5]*(2*Mappxyyz + Sqrt[3]*Mapy2xz + Mapz2x2xz))/4, k == 4 && m == -3}, {(-3*(4*Eappyz - 4*Eapxz - 3*Eapy2 + 3*Eapz2x2 + 2*Sqrt[3]*Mapz2x2y2))/(4*Sqrt[10]), k == 4 && (m == -2 || m == 2)}, {(-3*(2*Mappxyyz + Sqrt[3]*Mapy2xz - 7*Mapz2x2xz))/(4*Sqrt[5]), k == 4 && m == -1}, {(3*(4*Eappxy - 16*Eappyz - 16*Eapxz + 9*Eapy2 + 19*Eapz2x2 - 10*Sqrt[3]*Mapz2x2y2))/40, k == 4 && m == 0}, {(3*(2*Mappxyyz + Sqrt[3]*Mapy2xz - 7*Mapz2x2xz))/(4*Sqrt[5]), k == 4 && m == 1}}, (3*Sqrt[7/5]*(2*Mappxyyz + Sqrt[3]*Mapy2xz + Mapz2x2xz))/4] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{0, 0, (1/5)*(Eappxy + Eappyz + Eapxz + Eapy2 + Eapz2x2)} , |
| | {2, 0, (1/2)*((-2)*(Eappxy) + Eappyz + Eapxz + (-1)*(Eapy2) + Eapz2x2 + (-2)*((sqrt(3))*(Mapz2x2y2)))} , |
| | {2,-1, (1/(sqrt(2)))*((sqrt(3))*(Mappxyyz) + (-2)*(Mapy2xz))} , |
| | {2, 1, (-1)*((sqrt(3/2))*(Mappxyyz)) + (sqrt(2))*(Mapy2xz)} , |
| | {2,-2, (1/4)*((-1)*((sqrt(6))*(Eappyz)) + (sqrt(6))*(Eapxz) + (-1)*((sqrt(6))*(Eapy2)) + (sqrt(6))*(Eapz2x2) + (2)*((sqrt(2))*(Mapz2x2y2)))} , |
| | {2, 2, (1/4)*((-1)*((sqrt(6))*(Eappyz)) + (sqrt(6))*(Eapxz) + (-1)*((sqrt(6))*(Eapy2)) + (sqrt(6))*(Eapz2x2) + (2)*((sqrt(2))*(Mapz2x2y2)))} , |
| | {4, 0, (3/40)*((4)*(Eappxy) + (-16)*(Eappyz) + (-16)*(Eapxz) + (9)*(Eapy2) + (19)*(Eapz2x2) + (-10)*((sqrt(3))*(Mapz2x2y2)))} , |
| | {4,-1, (-3/4)*((1/(sqrt(5)))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + (-7)*(Mapz2x2xz)))} , |
| | {4, 1, (3/4)*((1/(sqrt(5)))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + (-7)*(Mapz2x2xz)))} , |
| | {4,-2, (-3/4)*((1/(sqrt(10)))*((4)*(Eappyz) + (-4)*(Eapxz) + (-3)*(Eapy2) + (3)*(Eapz2x2) + (2)*((sqrt(3))*(Mapz2x2y2))))} , |
| | {4, 2, (-3/4)*((1/(sqrt(10)))*((4)*(Eappyz) + (-4)*(Eapxz) + (-3)*(Eapy2) + (3)*(Eapz2x2) + (2)*((sqrt(3))*(Mapz2x2y2))))} , |
| | {4,-3, (-3/4)*((sqrt(7/5))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + Mapz2x2xz))} , |
| | {4, 3, (3/4)*((sqrt(7/5))*((2)*(Mappxyyz) + (sqrt(3))*(Mapy2xz) + Mapz2x2xz))} , |
| | {4,-4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapy2) + (-1)*(Eapz2x2) + (-2)*((sqrt(3))*(Mapz2x2y2))))} , |
| | {4, 4, (-3/8)*((sqrt(7/10))*((4)*(Eappxy) + (-3)*(Eapy2) + (-1)*(Eapz2x2) + (-2)*((sqrt(3))*(Mapz2x2y2))))} } |
| | |
| | </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{1}{8} \left(4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right) $|$ \frac{1}{4} \left(2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) $|$ \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) $|$ \frac{1}{4} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) $|$ \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right) $| |
| | ^$ {Y_{-1}^{(2)}} $|$ \frac{1}{4} \left(2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) $|$ \frac{\text{Eappyz}+\text{Eapxz}}{2} $|$ \frac{1}{4} \left(\sqrt{6} \text{Mapz2x2xz}-\sqrt{2} \text{Mapy2xz}\right) $|$ \frac{\text{Eappyz}-\text{Eapxz}}{2} $|$ \frac{1}{4} \left(-2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) $| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) $|$ \frac{1}{4} \left(\sqrt{6} \text{Mapz2x2xz}-\sqrt{2} \text{Mapy2xz}\right) $|$ \frac{1}{4} \left(\text{Eapy2}+3 \text{Eapz2x2}-2 \sqrt{3} \text{Mapz2x2y2}\right) $|$ \frac{\text{Mapy2xz}-\sqrt{3} \text{Mapz2x2xz}}{2 \sqrt{2}} $|$ \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) $| |
| | ^$ {Y_{1}^{(2)}} $|$ \frac{1}{4} \left(2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) $|$ \frac{\text{Eappyz}-\text{Eapxz}}{2} $|$ \frac{\text{Mapy2xz}-\sqrt{3} \text{Mapz2x2xz}}{2 \sqrt{2}} $|$ \frac{\text{Eappyz}+\text{Eapxz}}{2} $|$ \frac{1}{4} \left(-2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) $| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{1}{8} \left(-4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right) $|$ \frac{1}{4} \left(-2 \text{Mappxyyz}-\sqrt{3} \text{Mapy2xz}-\text{Mapz2x2xz}\right) $|$ \frac{1}{8} \left(\sqrt{6} \text{Eapy2}-\sqrt{6} \text{Eapz2x2}-2 \sqrt{2} \text{Mapz2x2y2}\right) $|$ \frac{1}{4} \left(-2 \text{Mappxyyz}+\sqrt{3} \text{Mapy2xz}+\text{Mapz2x2xz}\right) $|$ \frac{1}{8} \left(4 \text{Eappxy}+3 \text{Eapy2}+\text{Eapz2x2}+2 \sqrt{3} \text{Mapz2x2y2}\right) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{z^2-x^2} $ ^ $ d_{3y^2-r^2} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ |
| | ^$ d_{z^2-x^2} $|$ \text{Eapz2x2} $|$ \text{Mapz2x2y2} $|$ 0 $|$ 0 $|$ \text{Mapz2x2xz} $| |
| | ^$ d_{3y^2-r^2} $|$ \text{Mapz2x2y2} $|$ \text{Eapy2} $|$ 0 $|$ 0 $|$ \text{Mapy2xz} $| |
| | ^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ \text{Eappxy} $|$ \text{Mappxyyz} $|$ 0 $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Mappxyyz} $|$ \text{Eappyz} $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ \text{Mapz2x2xz} $|$ \text{Mapy2xz} $|$ 0 $|$ 0 $|$ \text{Eapxz} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ d_{z^2-x^2} $|$ -\frac{1}{2 \sqrt{2}} $|$ 0 $|$ \frac{\sqrt{3}}{2} $|$ 0 $|$ -\frac{1}{2 \sqrt{2}} $| |
| | ^$ d_{3y^2-r^2} $|$ -\frac{\sqrt{\frac{3}{2}}}{2} $|$ 0 $|$ -\frac{1}{2} $|$ 0 $|$ -\frac{\sqrt{\frac{3}{2}}}{2} $| |
| | ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eapz2x2}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_2_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{15}{\pi }} \left(-2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{15}{\pi }} \left(-x^2+y^2+3 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eapy2}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_2_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{5}{\pi }} \left(6 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} \left(-3 x^2+3 y^2-3 z^2+1\right)$$ | ::: | |
| | ^ ^$$\text{Eappxy}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_2_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: | |
| | ^ ^$$\text{Eappyz}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_2_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | |
| | ^ ^$$\text{Eapxz}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_2_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for f orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{7} (\text{Eappxyz}+\text{Eappy3}+\text{Eappyz2x2}+\text{Eapx3}+\text{Eapxy2z2}+\text{Eapz3}+\text{Eapzx2y2}) & k=0\land m=0 \\ |
| | 0 & (k\neq 2\land k\neq 4\land k\neq 6)\lor (k\neq 4\land k\neq 6\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (k\neq 6\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4)\lor (m\neq -6\land m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5\land m\neq 6) \\ |
| | \frac{5}{28} \left(-\sqrt{6} \text{Eappy3}+\sqrt{6} \text{Eapx3}+\sqrt{10} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2}-2 \text{Mapz3zx2y2})\right) & k=2\land (m=-2\lor m=2) \\ |
| | -\frac{5}{56} \left(4 \sqrt{10} \text{Mappxyzy3}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapz3x3}+\sqrt{10} \text{Mapz3xy2z2}+5 \sqrt{6} \text{Mapzx2y2xy2z2}\right) & k=2\land m=-1 \\ |
| | -\frac{5}{14} \left(\text{Eappy3}+\text{Eapx3}-2 \text{Eapz3}+\sqrt{15} \text{Mappy3yz2x2}-\sqrt{15} \text{Mapx3xy2z2}\right) & k=2\land m=0 \\ |
| | \frac{5}{56} \left(4 \sqrt{10} \text{Mappxyzy3}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapz3x3}+\sqrt{10} \text{Mapz3xy2z2}+5 \sqrt{6} \text{Mapzx2y2xy2z2}\right) & k=2\land m=1 \\ |
| | -\frac{3 \left(4 \sqrt{5} \text{Eappxyz}-3 \sqrt{5} \text{Eappy3}+3 \sqrt{5} \text{Eappyz2x2}-3 \sqrt{5} \text{Eapx3}+3 \sqrt{5} \text{Eapxy2z2}-4 \sqrt{5} \text{Eapzx2y2}+2 \sqrt{3} \text{Mappy3yz2x2}-2 \sqrt{3} \text{Mapx3xy2z2}\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\ |
| | \frac{3 \sqrt{3} \text{Mappxyzy3}-3 \left(\sqrt{5} \text{Mappxyzyz2x2}+\sqrt{3} \text{Mapx3zx2y2}+3 \sqrt{5} \text{Mapz3x3}-3 \sqrt{3} \text{Mapz3xy2z2}+\sqrt{5} \text{Mapzx2y2xy2z2}\right)}{4 \sqrt{7}} & k=4\land m=-3 \\ |
| | \frac{3}{56} \left(3 \sqrt{10} \text{Eappy3}-7 \sqrt{10} \text{Eappyz2x2}-3 \sqrt{10} \text{Eapx3}+7 \sqrt{10} \text{Eapxy2z2}+2 \sqrt{6} \text{Mappy3yz2x2}+2 \sqrt{6} \text{Mapx3xy2z2}-4 \sqrt{6} \text{Mapz3zx2y2}\right) & k=4\land (m=-2\lor m=2) \\ |
| | \frac{3}{28} \left(\sqrt{3} \text{Mappxyzy3}+7 \sqrt{5} \text{Mappxyzyz2x2}-9 \sqrt{3} \text{Mapx3zx2y2}-3 \sqrt{5} \text{Mapz3x3}-5 \sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapzx2y2xy2z2}\right) & k=4\land m=-1 \\ |
| | -\frac{3}{56} \left(28 \text{Eappxyz}-9 \text{Eappy3}-7 \text{Eappyz2x2}-9 \text{Eapx3}-7 \text{Eapxy2z2}-24 \text{Eapz3}+28 \text{Eapzx2y2}-2 \sqrt{15} \text{Mappy3yz2x2}+2 \sqrt{15} \text{Mapx3xy2z2}\right) & k=4\land m=0 \\ |
| | -\frac{3}{28} \left(\sqrt{3} \text{Mappxyzy3}+7 \sqrt{5} \text{Mappxyzyz2x2}-9 \sqrt{3} \text{Mapx3zx2y2}-3 \sqrt{5} \text{Mapz3x3}-5 \sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapzx2y2xy2z2}\right) & k=4\land m=1 \\ |
| | \frac{3 \left(-\sqrt{3} \text{Mappxyzy3}+\sqrt{5} \text{Mappxyzyz2x2}+\sqrt{3} \text{Mapx3zx2y2}+3 \sqrt{5} \text{Mapz3x3}-3 \sqrt{3} \text{Mapz3xy2z2}+\sqrt{5} \text{Mapzx2y2xy2z2}\right)}{4 \sqrt{7}} & k=4\land m=3 \\ |
| | -\frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eappy3}+3 \sqrt{3} \text{Eappyz2x2}-5 \sqrt{3} \text{Eapx3}-3 \sqrt{3} \text{Eapxy2z2}+6 \sqrt{5} \text{Mappy3yz2x2}+6 \sqrt{5} \text{Mapx3xy2z2}\right) & k=6\land (m=-6\lor m=6) \\ |
| | \frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}\right) & k=6\land m=-5 \\ |
| | -\frac{13 \left(24 \text{Eappxyz}+15 \text{Eappy3}-15 \text{Eappyz2x2}+15 \text{Eapx3}-15 \text{Eapxy2z2}-24 \text{Eapzx2y2}-2 \sqrt{15} \text{Mappy3yz2x2}+2 \sqrt{15} \text{Mapx3xy2z2}\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ |
| | \frac{13 \left(9 \text{Mappxyzy3}-3 \sqrt{15} \text{Mappxyzyz2x2}-9 \text{Mapx3zx2y2}+2 \sqrt{15} \text{Mapz3x3}-6 \text{Mapz3xy2z2}-3 \sqrt{15} \text{Mapzx2y2xy2z2}\right)}{40 \sqrt{7}} & k=6\land m=-3 \\ |
| | -\frac{13 \left(5 \sqrt{15} \text{Eappy3}+3 \sqrt{15} \text{Eappyz2x2}-5 \sqrt{15} \text{Eapx3}-3 \sqrt{15} \text{Eapxy2z2}-34 \text{Mappy3yz2x2}-34 \text{Mapx3xy2z2}-64 \text{Mapz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\ |
| | \frac{13}{280} \left(\sqrt{70} \text{Mappxyzy3}-3 \sqrt{42} \text{Mappxyzyz2x2}+2 \sqrt{70} \text{Mapx3zx2y2}-5 \sqrt{42} \text{Mapz3x3}-5 \sqrt{70} \text{Mapz3xy2z2}+2 \sqrt{42} \text{Mapzx2y2xy2z2}\right) & k=6\land m=-1 \\ |
| | \frac{13}{560} \left(24 \text{Eappxyz}-25 \text{Eappy3}-39 \text{Eappyz2x2}-25 \text{Eapx3}-39 \text{Eapxy2z2}+80 \text{Eapz3}+24 \text{Eapzx2y2}+14 \sqrt{15} \text{Mappy3yz2x2}-14 \sqrt{15} \text{Mapx3xy2z2}\right) & k=6\land m=0 \\ |
| | -\frac{13}{280} \left(\sqrt{70} \text{Mappxyzy3}-3 \sqrt{42} \text{Mappxyzyz2x2}+2 \sqrt{70} \text{Mapx3zx2y2}-5 \sqrt{42} \text{Mapz3x3}-5 \sqrt{70} \text{Mapz3xy2z2}+2 \sqrt{42} \text{Mapzx2y2xy2z2}\right) & k=6\land m=1 \\ |
| | -\frac{13 \left(9 \text{Mappxyzy3}-3 \sqrt{15} \text{Mappxyzyz2x2}-9 \text{Mapx3zx2y2}+2 \sqrt{15} \text{Mapz3x3}-6 \text{Mapz3xy2z2}-3 \sqrt{15} \text{Mapzx2y2xy2z2}\right)}{40 \sqrt{7}} & k=6\land m=3 \\ |
| | -\frac{13}{40} \sqrt{\frac{11}{7}} \left(\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}\right) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eappxyz + Eappy3 + Eappyz2x2 + Eapx3 + Eapxy2z2 + Eapz3 + Eapzx2y2)/7, k == 0 && m == 0}, {0, (k != 2 && k != 4 && k != 6) || (k != 4 && k != 6 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (k != 6 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4) || (m != -6 && m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5 && m != 6)}, {(5*(-(Sqrt[6]*Eappy3) + Sqrt[6]*Eapx3 + Sqrt[10]*(Mappy3yz2x2 + Mapx3xy2z2 - 2*Mapz3zx2y2)))/28, k == 2 && (m == -2 || m == 2)}, {(-5*(4*Sqrt[10]*Mappxyzy3 - Sqrt[10]*Mapx3zx2y2 + Sqrt[6]*Mapz3x3 + Sqrt[10]*Mapz3xy2z2 + 5*Sqrt[6]*Mapzx2y2xy2z2))/56, k == 2 && m == -1}, {(-5*(Eappy3 + Eapx3 - 2*Eapz3 + Sqrt[15]*Mappy3yz2x2 - Sqrt[15]*Mapx3xy2z2))/14, k == 2 && m == 0}, {(5*(4*Sqrt[10]*Mappxyzy3 - Sqrt[10]*Mapx3zx2y2 + Sqrt[6]*Mapz3x3 + Sqrt[10]*Mapz3xy2z2 + 5*Sqrt[6]*Mapzx2y2xy2z2))/56, k == 2 && m == 1}, {(-3*(4*Sqrt[5]*Eappxyz - 3*Sqrt[5]*Eappy3 + 3*Sqrt[5]*Eappyz2x2 - 3*Sqrt[5]*Eapx3 + 3*Sqrt[5]*Eapxy2z2 - 4*Sqrt[5]*Eapzx2y2 + 2*Sqrt[3]*Mappy3yz2x2 - 2*Sqrt[3]*Mapx3xy2z2))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(3*Sqrt[3]*Mappxyzy3 - 3*(Sqrt[5]*Mappxyzyz2x2 + Sqrt[3]*Mapx3zx2y2 + 3*Sqrt[5]*Mapz3x3 - 3*Sqrt[3]*Mapz3xy2z2 + Sqrt[5]*Mapzx2y2xy2z2))/(4*Sqrt[7]), k == 4 && m == -3}, {(3*(3*Sqrt[10]*Eappy3 - 7*Sqrt[10]*Eappyz2x2 - 3*Sqrt[10]*Eapx3 + 7*Sqrt[10]*Eapxy2z2 + 2*Sqrt[6]*Mappy3yz2x2 + 2*Sqrt[6]*Mapx3xy2z2 - 4*Sqrt[6]*Mapz3zx2y2))/56, k == 4 && (m == -2 || m == 2)}, {(3*(Sqrt[3]*Mappxyzy3 + 7*Sqrt[5]*Mappxyzyz2x2 - 9*Sqrt[3]*Mapx3zx2y2 - 3*Sqrt[5]*Mapz3x3 - 5*Sqrt[3]*Mapz3xy2z2 - Sqrt[5]*Mapzx2y2xy2z2))/28, k == 4 && m == -1}, {(-3*(28*Eappxyz - 9*Eappy3 - 7*Eappyz2x2 - 9*Eapx3 - 7*Eapxy2z2 - 24*Eapz3 + 28*Eapzx2y2 - 2*Sqrt[15]*Mappy3yz2x2 + 2*Sqrt[15]*Mapx3xy2z2))/56, k == 4 && m == 0}, {(-3*(Sqrt[3]*Mappxyzy3 + 7*Sqrt[5]*Mappxyzyz2x2 - 9*Sqrt[3]*Mapx3zx2y2 - 3*Sqrt[5]*Mapz3x3 - 5*Sqrt[3]*Mapz3xy2z2 - Sqrt[5]*Mapzx2y2xy2z2))/28, k == 4 && m == 1}, {(3*(-(Sqrt[3]*Mappxyzy3) + Sqrt[5]*Mappxyzyz2x2 + Sqrt[3]*Mapx3zx2y2 + 3*Sqrt[5]*Mapz3x3 - 3*Sqrt[3]*Mapz3xy2z2 + Sqrt[5]*Mapzx2y2xy2z2))/(4*Sqrt[7]), k == 4 && m == 3}, {(-13*Sqrt[11/7]*(5*Sqrt[3]*Eappy3 + 3*Sqrt[3]*Eappyz2x2 - 5*Sqrt[3]*Eapx3 - 3*Sqrt[3]*Eapxy2z2 + 6*Sqrt[5]*Mappy3yz2x2 + 6*Sqrt[5]*Mapx3xy2z2))/160, k == 6 && (m == -6 || m == 6)}, {(13*Sqrt[11/7]*(Sqrt[15]*Mappxyzy3 + 3*Mappxyzyz2x2 + Sqrt[15]*Mapx3zx2y2 - 3*Mapzx2y2xy2z2))/40, k == 6 && m == -5}, {(-13*(24*Eappxyz + 15*Eappy3 - 15*Eappyz2x2 + 15*Eapx3 - 15*Eapxy2z2 - 24*Eapzx2y2 - 2*Sqrt[15]*Mappy3yz2x2 + 2*Sqrt[15]*Mapx3xy2z2))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(13*(9*Mappxyzy3 - 3*Sqrt[15]*Mappxyzyz2x2 - 9*Mapx3zx2y2 + 2*Sqrt[15]*Mapz3x3 - 6*Mapz3xy2z2 - 3*Sqrt[15]*Mapzx2y2xy2z2))/(40*Sqrt[7]), k == 6 && m == -3}, {(-13*(5*Sqrt[15]*Eappy3 + 3*Sqrt[15]*Eappyz2x2 - 5*Sqrt[15]*Eapx3 - 3*Sqrt[15]*Eapxy2z2 - 34*Mappy3yz2x2 - 34*Mapx3xy2z2 - 64*Mapz3zx2y2))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {(13*(Sqrt[70]*Mappxyzy3 - 3*Sqrt[42]*Mappxyzyz2x2 + 2*Sqrt[70]*Mapx3zx2y2 - 5*Sqrt[42]*Mapz3x3 - 5*Sqrt[70]*Mapz3xy2z2 + 2*Sqrt[42]*Mapzx2y2xy2z2))/280, k == 6 && m == -1}, {(13*(24*Eappxyz - 25*Eappy3 - 39*Eappyz2x2 - 25*Eapx3 - 39*Eapxy2z2 + 80*Eapz3 + 24*Eapzx2y2 + 14*Sqrt[15]*Mappy3yz2x2 - 14*Sqrt[15]*Mapx3xy2z2))/560, k == 6 && m == 0}, {(-13*(Sqrt[70]*Mappxyzy3 - 3*Sqrt[42]*Mappxyzyz2x2 + 2*Sqrt[70]*Mapx3zx2y2 - 5*Sqrt[42]*Mapz3x3 - 5*Sqrt[70]*Mapz3xy2z2 + 2*Sqrt[42]*Mapzx2y2xy2z2))/280, k == 6 && m == 1}, {(-13*(9*Mappxyzy3 - 3*Sqrt[15]*Mappxyzyz2x2 - 9*Mapx3zx2y2 + 2*Sqrt[15]*Mapz3x3 - 6*Mapz3xy2z2 - 3*Sqrt[15]*Mapzx2y2xy2z2))/(40*Sqrt[7]), k == 6 && m == 3}}, (-13*Sqrt[11/7]*(Sqrt[15]*Mappxyzy3 + 3*Mappxyzyz2x2 + Sqrt[15]*Mapx3zx2y2 - 3*Mapzx2y2xy2z2))/40] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{0, 0, (1/7)*(Eappxyz + Eappy3 + Eappyz2x2 + Eapx3 + Eapxy2z2 + Eapz3 + Eapzx2y2)} , |
| | {2, 0, (-5/14)*(Eappy3 + Eapx3 + (-2)*(Eapz3) + (sqrt(15))*(Mappy3yz2x2) + (-1)*((sqrt(15))*(Mapx3xy2z2)))} , |
| | {2,-1, (-5/56)*((4)*((sqrt(10))*(Mappxyzy3)) + (-1)*((sqrt(10))*(Mapx3zx2y2)) + (sqrt(6))*(Mapz3x3) + (sqrt(10))*(Mapz3xy2z2) + (5)*((sqrt(6))*(Mapzx2y2xy2z2)))} , |
| | {2, 1, (5/56)*((4)*((sqrt(10))*(Mappxyzy3)) + (-1)*((sqrt(10))*(Mapx3zx2y2)) + (sqrt(6))*(Mapz3x3) + (sqrt(10))*(Mapz3xy2z2) + (5)*((sqrt(6))*(Mapzx2y2xy2z2)))} , |
| | {2,-2, (5/28)*((-1)*((sqrt(6))*(Eappy3)) + (sqrt(6))*(Eapx3) + (sqrt(10))*(Mappy3yz2x2 + Mapx3xy2z2 + (-2)*(Mapz3zx2y2)))} , |
| | {2, 2, (5/28)*((-1)*((sqrt(6))*(Eappy3)) + (sqrt(6))*(Eapx3) + (sqrt(10))*(Mappy3yz2x2 + Mapx3xy2z2 + (-2)*(Mapz3zx2y2)))} , |
| | {4, 0, (-3/56)*((28)*(Eappxyz) + (-9)*(Eappy3) + (-7)*(Eappyz2x2) + (-9)*(Eapx3) + (-7)*(Eapxy2z2) + (-24)*(Eapz3) + (28)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappy3yz2x2)) + (2)*((sqrt(15))*(Mapx3xy2z2)))} , |
| | {4, 1, (-3/28)*((sqrt(3))*(Mappxyzy3) + (7)*((sqrt(5))*(Mappxyzyz2x2)) + (-9)*((sqrt(3))*(Mapx3zx2y2)) + (-3)*((sqrt(5))*(Mapz3x3)) + (-5)*((sqrt(3))*(Mapz3xy2z2)) + (-1)*((sqrt(5))*(Mapzx2y2xy2z2)))} , |
| | {4,-1, (3/28)*((sqrt(3))*(Mappxyzy3) + (7)*((sqrt(5))*(Mappxyzyz2x2)) + (-9)*((sqrt(3))*(Mapx3zx2y2)) + (-3)*((sqrt(5))*(Mapz3x3)) + (-5)*((sqrt(3))*(Mapz3xy2z2)) + (-1)*((sqrt(5))*(Mapzx2y2xy2z2)))} , |
| | {4,-2, (3/56)*((3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (-3)*((sqrt(10))*(Eapx3)) + (7)*((sqrt(10))*(Eapxy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (2)*((sqrt(6))*(Mapx3xy2z2)) + (-4)*((sqrt(6))*(Mapz3zx2y2)))} , |
| | {4, 2, (3/56)*((3)*((sqrt(10))*(Eappy3)) + (-7)*((sqrt(10))*(Eappyz2x2)) + (-3)*((sqrt(10))*(Eapx3)) + (7)*((sqrt(10))*(Eapxy2z2)) + (2)*((sqrt(6))*(Mappy3yz2x2)) + (2)*((sqrt(6))*(Mapx3xy2z2)) + (-4)*((sqrt(6))*(Mapz3zx2y2)))} , |
| | {4,-3, (1/4)*((1/(sqrt(7)))*((3)*((sqrt(3))*(Mappxyzy3)) + (-3)*((sqrt(5))*(Mappxyzyz2x2) + (sqrt(3))*(Mapx3zx2y2) + (3)*((sqrt(5))*(Mapz3x3)) + (-3)*((sqrt(3))*(Mapz3xy2z2)) + (sqrt(5))*(Mapzx2y2xy2z2))))} , |
| | {4, 3, (3/4)*((1/(sqrt(7)))*((-1)*((sqrt(3))*(Mappxyzy3)) + (sqrt(5))*(Mappxyzyz2x2) + (sqrt(3))*(Mapx3zx2y2) + (3)*((sqrt(5))*(Mapz3x3)) + (-3)*((sqrt(3))*(Mapz3xy2z2)) + (sqrt(5))*(Mapzx2y2xy2z2)))} , |
| | {4,-4, (-3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Eappxyz)) + (-3)*((sqrt(5))*(Eappy3)) + (3)*((sqrt(5))*(Eappyz2x2)) + (-3)*((sqrt(5))*(Eapx3)) + (3)*((sqrt(5))*(Eapxy2z2)) + (-4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappy3yz2x2)) + (-2)*((sqrt(3))*(Mapx3xy2z2))))} , |
| | {4, 4, (-3/8)*((1/(sqrt(14)))*((4)*((sqrt(5))*(Eappxyz)) + (-3)*((sqrt(5))*(Eappy3)) + (3)*((sqrt(5))*(Eappyz2x2)) + (-3)*((sqrt(5))*(Eapx3)) + (3)*((sqrt(5))*(Eapxy2z2)) + (-4)*((sqrt(5))*(Eapzx2y2)) + (2)*((sqrt(3))*(Mappy3yz2x2)) + (-2)*((sqrt(3))*(Mapx3xy2z2))))} , |
| | {6, 0, (13/560)*((24)*(Eappxyz) + (-25)*(Eappy3) + (-39)*(Eappyz2x2) + (-25)*(Eapx3) + (-39)*(Eapxy2z2) + (80)*(Eapz3) + (24)*(Eapzx2y2) + (14)*((sqrt(15))*(Mappy3yz2x2)) + (-14)*((sqrt(15))*(Mapx3xy2z2)))} , |
| | {6, 1, (-13/280)*((sqrt(70))*(Mappxyzy3) + (-3)*((sqrt(42))*(Mappxyzyz2x2)) + (2)*((sqrt(70))*(Mapx3zx2y2)) + (-5)*((sqrt(42))*(Mapz3x3)) + (-5)*((sqrt(70))*(Mapz3xy2z2)) + (2)*((sqrt(42))*(Mapzx2y2xy2z2)))} , |
| | {6,-1, (13/280)*((sqrt(70))*(Mappxyzy3) + (-3)*((sqrt(42))*(Mappxyzyz2x2)) + (2)*((sqrt(70))*(Mapx3zx2y2)) + (-5)*((sqrt(42))*(Mapz3x3)) + (-5)*((sqrt(70))*(Mapz3xy2z2)) + (2)*((sqrt(42))*(Mapzx2y2xy2z2)))} , |
| | {6,-2, (-13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappy3)) + (3)*((sqrt(15))*(Eappyz2x2)) + (-5)*((sqrt(15))*(Eapx3)) + (-3)*((sqrt(15))*(Eapxy2z2)) + (-34)*(Mappy3yz2x2) + (-34)*(Mapx3xy2z2) + (-64)*(Mapz3zx2y2)))} , |
| | {6, 2, (-13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eappy3)) + (3)*((sqrt(15))*(Eappyz2x2)) + (-5)*((sqrt(15))*(Eapx3)) + (-3)*((sqrt(15))*(Eapxy2z2)) + (-34)*(Mappy3yz2x2) + (-34)*(Mapx3xy2z2) + (-64)*(Mapz3zx2y2)))} , |
| | {6, 3, (-13/40)*((1/(sqrt(7)))*((9)*(Mappxyzy3) + (-3)*((sqrt(15))*(Mappxyzyz2x2)) + (-9)*(Mapx3zx2y2) + (2)*((sqrt(15))*(Mapz3x3)) + (-6)*(Mapz3xy2z2) + (-3)*((sqrt(15))*(Mapzx2y2xy2z2))))} , |
| | {6,-3, (13/40)*((1/(sqrt(7)))*((9)*(Mappxyzy3) + (-3)*((sqrt(15))*(Mappxyzyz2x2)) + (-9)*(Mapx3zx2y2) + (2)*((sqrt(15))*(Mapz3x3)) + (-6)*(Mapz3xy2z2) + (-3)*((sqrt(15))*(Mapzx2y2xy2z2))))} , |
| | {6,-4, (-13/80)*((1/(sqrt(14)))*((24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (15)*(Eapx3) + (-15)*(Eapxy2z2) + (-24)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappy3yz2x2)) + (2)*((sqrt(15))*(Mapx3xy2z2))))} , |
| | {6, 4, (-13/80)*((1/(sqrt(14)))*((24)*(Eappxyz) + (15)*(Eappy3) + (-15)*(Eappyz2x2) + (15)*(Eapx3) + (-15)*(Eapxy2z2) + (-24)*(Eapzx2y2) + (-2)*((sqrt(15))*(Mappy3yz2x2)) + (2)*((sqrt(15))*(Mapx3xy2z2))))} , |
| | {6, 5, (-13/40)*((sqrt(11/7))*((sqrt(15))*(Mappxyzy3) + (3)*(Mappxyzyz2x2) + (sqrt(15))*(Mapx3zx2y2) + (-3)*(Mapzx2y2xy2z2)))} , |
| | {6,-5, (13/40)*((sqrt(11/7))*((sqrt(15))*(Mappxyzy3) + (3)*(Mappxyzyz2x2) + (sqrt(15))*(Mapx3zx2y2) + (-3)*(Mapzx2y2xy2z2)))} , |
| | {6,-6, (-13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappy3)) + (3)*((sqrt(3))*(Eappyz2x2)) + (-5)*((sqrt(3))*(Eapx3)) + (-3)*((sqrt(3))*(Eapxy2z2)) + (6)*((sqrt(5))*(Mappy3yz2x2)) + (6)*((sqrt(5))*(Mapx3xy2z2))))} , |
| | {6, 6, (-13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eappy3)) + (3)*((sqrt(3))*(Eappyz2x2)) + (-5)*((sqrt(3))*(Eapx3)) + (-3)*((sqrt(3))*(Eapxy2z2)) + (6)*((sqrt(5))*(Mappy3yz2x2)) + (6)*((sqrt(5))*(Mapx3xy2z2))))} } |
| | |
| | </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{Eappy3}+3 \text{Eappyz2x2}+5 \text{Eapx3}+3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mapx3xy2z2})\right) $|$ \frac{1}{8} \left(-\sqrt{10} \text{Mappxyzy3}-\sqrt{6} \text{Mappxyzyz2x2}+\sqrt{10} \text{Mapx3zx2y2}-\sqrt{6} \text{Mapzx2y2xy2z2}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $|$ \frac{1}{4} \left(\sqrt{5} \text{Mapz3x3}-\sqrt{3} \text{Mapz3xy2z2}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) $|$ \frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} $|$ \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}-5 \text{Eapx3}-3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $| |
| | ^$ {Y_{-2}^{(3)}} $|$ \frac{1}{8} \left(-\sqrt{10} \text{Mappxyzy3}-\sqrt{6} \text{Mappxyzyz2x2}+\sqrt{10} \text{Mapx3zx2y2}-\sqrt{6} \text{Mapzx2y2xy2z2}\right) $|$ \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} $|$ \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} $|$ -\frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} $| |
| | ^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $|$ \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}+3 \text{Eapx3}+5 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mapx3xy2z2}-\text{Mappy3yz2x2})\right) $|$ \frac{1}{4} \left(-\sqrt{3} \text{Mapz3x3}-\sqrt{5} \text{Mapz3xy2z2}\right) $|$ \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}-3 \text{Eapx3}-5 \text{Eapxy2z2}-2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $|$ \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) $| |
| | ^$ {Y_{0}^{(3)}} $|$ \frac{1}{4} \left(\sqrt{5} \text{Mapz3x3}-\sqrt{3} \text{Mapz3xy2z2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{4} \left(-\sqrt{3} \text{Mapz3x3}-\sqrt{5} \text{Mapz3xy2z2}\right) $|$ \text{Eapz3} $|$ \frac{1}{4} \left(\sqrt{3} \text{Mapz3x3}+\sqrt{5} \text{Mapz3xy2z2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{4} \left(\sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapz3x3}\right) $| |
| | ^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) $|$ \frac{1}{8} \left(-\sqrt{6} \text{Mappxyzy3}+\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}-3 \text{Eapx3}-5 \text{Eapxy2z2}-2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $|$ \frac{1}{4} \left(\sqrt{3} \text{Mapz3x3}+\sqrt{5} \text{Mapz3xy2z2}\right) $|$ \frac{1}{16} \left(3 \text{Eappy3}+5 \text{Eappyz2x2}+3 \text{Eapx3}+5 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mapx3xy2z2}-\text{Mappy3yz2x2})\right) $|$ \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $| |
| | ^$ {Y_{2}^{(3)}} $|$ \frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} $|$ \frac{\text{Eapzx2y2}-\text{Eappxyz}}{2} $|$ \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}-\sqrt{6} \text{Mapx3zx2y2}-\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{\text{Mapz3zx2y2}}{\sqrt{2}} $|$ \frac{1}{8} \left(\sqrt{6} \text{Mappxyzy3}-\sqrt{10} \text{Mappxyzyz2x2}+\sqrt{6} \text{Mapx3zx2y2}+\sqrt{10} \text{Mapzx2y2xy2z2}\right) $|$ \frac{\text{Eappxyz}+\text{Eapzx2y2}}{2} $|$ \frac{1}{8} \left(\sqrt{10} \text{Mappxyzy3}+\sqrt{6} \text{Mappxyzyz2x2}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapzx2y2xy2z2}\right) $| |
| | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}-5 \text{Eapx3}-3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $|$ -\frac{\sqrt{15} \text{Mappxyzy3}+3 \text{Mappxyzyz2x2}+\sqrt{15} \text{Mapx3zx2y2}-3 \text{Mapzx2y2xy2z2}}{4 \sqrt{6}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}+\sqrt{15} \text{Eapx3}-\sqrt{15} \text{Eapxy2z2}-2 \text{Mappy3yz2x2}+2 \text{Mapx3xy2z2}\right) $|$ \frac{1}{4} \left(\sqrt{3} \text{Mapz3xy2z2}-\sqrt{5} \text{Mapz3x3}\right) $|$ \frac{1}{16} \left(\sqrt{15} \text{Eappy3}-\sqrt{15} \text{Eappyz2x2}-\sqrt{15} \text{Eapx3}+\sqrt{15} \text{Eapxy2z2}-2 (\text{Mappy3yz2x2}+\text{Mapx3xy2z2})\right) $|$ \frac{1}{8} \left(\sqrt{10} \text{Mappxyzy3}+\sqrt{6} \text{Mappxyzyz2x2}-\sqrt{10} \text{Mapx3zx2y2}+\sqrt{6} \text{Mapzx2y2xy2z2}\right) $|$ \frac{1}{16} \left(5 \text{Eappy3}+3 \text{Eappyz2x2}+5 \text{Eapx3}+3 \text{Eapxy2z2}+2 \sqrt{15} (\text{Mappy3yz2x2}-\text{Mapx3xy2z2})\right) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ |
| | ^$ f_{\text{xyz}} $|$ \text{Eappxyz} $|$ 0 $|$ 0 $|$ \text{Mappxyzy3} $|$ 0 $|$ 0 $|$ \text{Mappxyzyz2x2} $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ \text{Eapz3} $|$ \text{Mapz3x3} $|$ 0 $|$ \text{Mapz3zx2y2} $|$ \text{Mapz3xy2z2} $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Mapz3x3} $|$ \text{Eapx3} $|$ 0 $|$ \text{Mapx3zx2y2} $|$ \text{Mapx3xy2z2} $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$ \text{Mappxyzy3} $|$ 0 $|$ 0 $|$ \text{Eappy3} $|$ 0 $|$ 0 $|$ \text{Mappy3yz2x2} $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \text{Mapz3zx2y2} $|$ \text{Mapx3zx2y2} $|$ 0 $|$ \text{Eapzx2y2} $|$ \text{Mapzx2y2xy2z2} $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \text{Mapz3xy2z2} $|$ \text{Mapx3xy2z2} $|$ 0 $|$ \text{Mapzx2y2xy2z2} $|$ \text{Eapxy2z2} $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$ \text{Mappxyzyz2x2} $|$ 0 $|$ 0 $|$ \text{Mappy3yz2x2} $|$ 0 $|$ 0 $|$ \text{Eappyz2x2} $| |
| | |
| | |
| | ### |
| | |
| | </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_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| |
| | ^$ f_{y\left(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(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| |
| | ^$ 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} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eappxyz}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_3_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | |
| | ^ ^$$\text{Eapz3}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_3_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | |
| | ^ ^$$\text{Eapx3}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_3_3.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_y_orb_3_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\text{Eapzx2y2}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_3_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | |
| | ^ ^$$\text{Eapxy2z2}$$ | {{:physics_chemistry:pointgroup:cs_y_orb_3_6.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_y_orb_3_7.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </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** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 1\lor (m\neq -1\land m\neq 0\land m\neq 1) \\ |
| | -A(1,1) & k=1\land m=-1 \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(1,1) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 1 || (m != -1 && m != 0 && m != 1)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}}, A[1, 1]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{1, 0, A(1,0)} , |
| | {1,-1, (-1)*(A(1,1))} , |
| | {1, 1, A(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)}{\sqrt{3}} $|$ \frac{A(1,0)}{\sqrt{3}} $|$ \frac{A(1,1)}{\sqrt{3}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_z $ ^ $ p_x $ ^ $ p_y $ ^ |
| | ^$ \text{s} $|$ \frac{A(1,0)}{\sqrt{3}} $|$ -\sqrt{\frac{2}{3}} A(1,1) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for s-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 2\lor (m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2) \\ |
| | A(2,2) & k=2\land (m=-2\lor m=2) \\ |
| | -A(2,1) & k=2\land m=-1 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,1) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != -1 && m != 0 && m != 1 && m != 2)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {-A[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}}, A[2, 1]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{2, 0, A(2,0)} , |
| | {2,-1, (-1)*(A(2,1))} , |
| | {2, 1, A(2,1)} , |
| | {2,-2, A(2,2)} , |
| | {2, 2, A(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)}{\sqrt{5}} $|$ -\frac{A(2,1)}{\sqrt{5}} $|$ \frac{A(2,0)}{\sqrt{5}} $|$ \frac{A(2,1)}{\sqrt{5}} $|$ \frac{A(2,2)}{\sqrt{5}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{z^2-x^2} $ ^ $ d_{3y^2-r^2} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ |
| | ^$ \text{s} $|$ \frac{1}{10} \left(\sqrt{15} A(2,0)-\sqrt{10} A(2,2)\right) $|$ -\frac{A(2,0)+\sqrt{6} A(2,2)}{2 \sqrt{5}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} A(2,1) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for s-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 3\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3) \\ |
| | -A(3,3) & k=3\land m=-3 \\ |
| | A(3,2) & k=3\land (m=-2\lor m=2) \\ |
| | -A(3,1) & k=3\land m=-1 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,1) & k=3\land m=1 \\ |
| | A(3,3) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3)}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}}, A[3, 3]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{3, 0, A(3,0)} , |
| | {3,-1, (-1)*(A(3,1))} , |
| | {3, 1, A(3,1)} , |
| | {3,-2, A(3,2)} , |
| | {3, 2, A(3,2)} , |
| | {3,-3, (-1)*(A(3,3))} , |
| | {3, 3, A(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)}{\sqrt{7}} $|$ \frac{A(3,2)}{\sqrt{7}} $|$ -\frac{A(3,1)}{\sqrt{7}} $|$ \frac{A(3,0)}{\sqrt{7}} $|$ \frac{A(3,1)}{\sqrt{7}} $|$ \frac{A(3,2)}{\sqrt{7}} $|$ \frac{A(3,3)}{\sqrt{7}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ |
| | ^$ \text{s} $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) $|$ 0 $|$ \sqrt{\frac{2}{7}} A(3,2) $|$ \frac{1}{14} \left(\sqrt{35} A(3,1)+\sqrt{21} A(3,3)\right) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 3\land (k\neq 1\lor (m\neq -1\land m\neq 0\land m\neq 1)))\lor (m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3) \\ |
| | -A(1,1) & k=1\land m=-1 \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(1,1) & k=1\land m=1 \\ |
| | -A(3,3) & k=3\land m=-3 \\ |
| | A(3,2) & k=3\land (m=-2\lor m=2) \\ |
| | -A(3,1) & k=3\land m=-1 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,1) & k=3\land m=1 \\ |
| | A(3,3) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || (m != -1 && m != 0 && m != 1))) || (m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1], k == 1 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}}, A[3, 3]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{1, 0, A(1,0)} , |
| | {1,-1, (-1)*(A(1,1))} , |
| | {1, 1, A(1,1)} , |
| | {3, 0, A(3,0)} , |
| | {3,-1, (-1)*(A(3,1))} , |
| | {3, 1, A(3,1)} , |
| | {3,-2, A(3,2)} , |
| | {3, 2, A(3,2)} , |
| | {3,-3, (-1)*(A(3,3))} , |
| | {3, 3, A(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}{35} \left(\sqrt{15} A(3,1)-7 \sqrt{10} A(1,1)\right) $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ \frac{A(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} A(3,1) $|$ -\frac{1}{7} \sqrt{6} A(3,2) $|$ -\frac{3}{7} A(3,3) $| |
| | ^$ {Y_{0}^{(1)}} $|$ \frac{1}{7} \sqrt{3} A(3,2) $|$ -\frac{7 A(1,1)+2 \sqrt{6} A(3,1)}{7 \sqrt{5}} $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ \frac{A(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} A(3,1) $|$ \frac{1}{7} \sqrt{3} A(3,2) $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{3}{7} A(3,3) $|$ -\frac{1}{7} \sqrt{6} A(3,2) $|$ \frac{3}{7} \sqrt{\frac{2}{5}} A(3,1)-\frac{A(1,1)}{\sqrt{15}} $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ \sqrt{\frac{2}{5}} A(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} A(3,1) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{z^2-x^2} $ ^ $ d_{3y^2-r^2} $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ |
| | ^$ p_z $|$ \frac{1}{70} \left(14 \sqrt{5} A(1,0)+9 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) $|$ -\frac{14 A(1,0)+9 A(3,0)}{14 \sqrt{15}}-\frac{3 A(3,2)}{7 \sqrt{2}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} A(1,1)-\frac{4}{7} \sqrt{\frac{3}{5}} A(3,1) $| |
| | ^$ p_x $|$ \sqrt{\frac{2}{5}} A(1,1)-\frac{1}{2} \sqrt{\frac{3}{5}} A(3,1)+\frac{3}{14} A(3,3) $|$ \frac{1}{210} \left(14 \sqrt{30} A(1,1)+9 \sqrt{5} A(3,1)+45 \sqrt{3} A(3,3)\right) $|$ 0 $|$ 0 $|$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right) $| |
| | ^$ p_y $|$ 0 $|$ 0 $|$ \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) $|$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ |
| | A(2,2) & k=2\land (m=-2\lor m=2) \\ |
| | -A(2,1) & k=2\land m=-1 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,1) & k=2\land m=1 \\ |
| | A(4,4) & k=4\land (m=-4\lor m=4) \\ |
| | -A(4,3) & k=4\land m=-3 \\ |
| | A(4,2) & k=4\land (m=-2\lor m=2) \\ |
| | -A(4,1) & k=4\land m=-1 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,1) & k=4\land m=1 \\ |
| | A(4,3) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {-A[2, 1], k == 2 && m == -1}, {A[2, 0], k == 2 && m == 0}, {A[2, 1], k == 2 && m == 1}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {-A[4, 3], k == 4 && m == -3}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {-A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {A[4, 1], k == 4 && m == 1}}, A[4, 3]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{2, 0, A(2,0)} , |
| | {2,-1, (-1)*(A(2,1))} , |
| | {2, 1, A(2,1)} , |
| | {2,-2, A(2,2)} , |
| | {2, 2, A(2,2)} , |
| | {4, 0, A(4,0)} , |
| | {4,-1, (-1)*(A(4,1))} , |
| | {4, 1, A(4,1)} , |
| | {4,-2, A(4,2)} , |
| | {4, 2, A(4,2)} , |
| | {4,-3, (-1)*(A(4,3))} , |
| | {4, 3, A(4,3)} , |
| | {4,-4, A(4,4)} , |
| | {4, 4, A(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)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $|$ \frac{A(4,1)}{3 \sqrt{7}}-\sqrt{\frac{6}{35}} A(2,1) $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ \frac{27 A(2,1)-5 \sqrt{30} A(4,1)}{45 \sqrt{7}} $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ -\frac{1}{3} A(4,3) $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$ -\frac{A(4,3)}{3 \sqrt{3}} $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ \frac{1}{105} \left(-6 \sqrt{42} A(2,1)-5 \sqrt{35} A(4,1)\right) $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ \frac{1}{105} \left(6 \sqrt{42} A(2,1)+5 \sqrt{35} A(4,1)\right) $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ \frac{A(4,3)}{3 \sqrt{3}} $| |
| | ^$ {Y_{1}^{(1)}} $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|$ \frac{1}{3} A(4,3) $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ \frac{5 \sqrt{30} A(4,1)-27 A(2,1)}{45 \sqrt{7}} $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ \sqrt{\frac{6}{35}} A(2,1)-\frac{A(4,1)}{3 \sqrt{7}} $|$ \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ |
| | ^$ p_z $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ \frac{1}{630} \left(54 \sqrt{14} A(2,1)+5 \sqrt{15} \left(3 \sqrt{7} A(4,1)-7 A(4,3)\right)\right) $|$ 0 $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $|$ \sqrt{\frac{6}{35}} A(2,1)+\frac{5 A(4,1)}{6 \sqrt{7}}+\frac{1}{6} A(4,3) $|$ 0 $| |
| | ^$ p_x $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} A(2,1)-\frac{2}{3} \sqrt{\frac{5}{21}} A(4,1) $|$ \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 $|$ \frac{1}{21} \left(\sqrt{7} A(4,1)-7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) $|$ \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_y $|$ \frac{1}{21} \left(\sqrt{7} A(4,1)+7 A(4,3)\right)-\sqrt{\frac{6}{35}} A(2,1) $|$ 0 $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ 0 $|$ 0 $|$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 1\land k\neq 3\land k\neq 5)\lor (k\neq 3\land k\neq 5\land m\neq -1\land m\neq 0\land m\neq 1)\lor (k\neq 5\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3)\lor (m\neq -5\land m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4\land m\neq 5) \\ |
| | -A(1,1) & k=1\land m=-1 \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(1,1) & k=1\land m=1 \\ |
| | -A(3,3) & k=3\land m=-3 \\ |
| | A(3,2) & k=3\land (m=-2\lor m=2) \\ |
| | -A(3,1) & k=3\land m=-1 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,1) & k=3\land m=1 \\ |
| | A(3,3) & k=3\land m=3 \\ |
| | -A(5,5) & k=5\land m=-5 \\ |
| | A(5,4) & k=5\land (m=-4\lor m=4) \\ |
| | -A(5,3) & k=5\land m=-3 \\ |
| | A(5,2) & k=5\land (m=-2\lor m=2) \\ |
| | -A(5,1) & k=5\land m=-1 \\ |
| | A(5,0) & k=5\land m=0 \\ |
| | A(5,1) & k=5\land m=1 \\ |
| | A(5,3) & k=5\land m=3 \\ |
| | A(5,5) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 1 && k != 3 && k != 5) || (k != 3 && k != 5 && m != -1 && m != 0 && m != 1) || (k != 5 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3) || (m != -5 && m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4 && m != 5)}, {-A[1, 1], k == 1 && m == -1}, {A[1, 0], k == 1 && m == 0}, {A[1, 1], k == 1 && m == 1}, {-A[3, 3], k == 3 && m == -3}, {A[3, 2], k == 3 && (m == -2 || m == 2)}, {-A[3, 1], k == 3 && m == -1}, {A[3, 0], k == 3 && m == 0}, {A[3, 1], k == 3 && m == 1}, {A[3, 3], k == 3 && m == 3}, {-A[5, 5], k == 5 && m == -5}, {A[5, 4], k == 5 && (m == -4 || m == 4)}, {-A[5, 3], k == 5 && m == -3}, {A[5, 2], k == 5 && (m == -2 || m == 2)}, {-A[5, 1], k == 5 && m == -1}, {A[5, 0], k == 5 && m == 0}, {A[5, 1], k == 5 && m == 1}, {A[5, 3], k == 5 && m == 3}}, A[5, 5]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Y.Quanty> |
| | |
| | Akm = {{1, 0, A(1,0)} , |
| | {1,-1, (-1)*(A(1,1))} , |
| | {1, 1, A(1,1)} , |
| | {3, 0, A(3,0)} , |
| | {3,-1, (-1)*(A(3,1))} , |
| | {3, 1, A(3,1)} , |
| | {3,-2, A(3,2)} , |
| | {3, 2, A(3,2)} , |
| | {3,-3, (-1)*(A(3,3))} , |
| | {3, 3, A(3,3)} , |
| | {5, 0, A(5,0)} , |
| | {5,-1, (-1)*(A(5,1))} , |
| | {5, 1, A(5,1)} , |
| | {5,-2, A(5,2)} , |
| | {5, 2, A(5,2)} , |
| | {5,-3, (-1)*(A(5,3))} , |
| | {5, 3, A(5,3)} , |
| | {5,-4, A(5,4)} , |
| | {5, 4, A(5,4)} , |
| | {5,-5, (-1)*(A(5,5))} , |
| | {5, 5, A(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)}} $|$ \frac{1}{231} \left(-33 \sqrt{21} A(1,1)+11 \sqrt{14} A(3,1)-\sqrt{35} A(5,1)\right) $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ \frac{A(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} A(3,1)+\frac{5 A(5,1)}{11 \sqrt{21}} $|$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $|$ \frac{5}{33} \sqrt{2} A(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3) $|$ \frac{1}{11} \sqrt{10} A(5,4) $|$ \frac{5}{11} \sqrt{\frac{2}{3}} A(5,5) $| |
| | ^$ {Y_{-1}^{(2)}} $|$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $|$ -\sqrt{\frac{2}{7}} A(1,1)-\frac{A(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} A(5,1) $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ \sqrt{\frac{3}{35}} A(1,1)-\frac{1}{3} \sqrt{\frac{2}{35}} A(3,1)-\frac{20 A(5,1)}{33 \sqrt{7}} $|$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $|$ -\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+28 A(5,3)\right) $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)-\frac{2}{33} \sqrt{5} A(5,3) $|$ \frac{1}{11} \sqrt{5} A(5,2) $|$ -\sqrt{\frac{6}{35}} A(1,1)-\frac{A(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} A(5,1) $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ \sqrt{\frac{6}{35}} A(1,1)+\frac{A(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} A(5,1) $|$ \frac{1}{11} \sqrt{5} A(5,2) $|$ \frac{2}{33} \sqrt{5} A(5,3)-\frac{1}{3} \sqrt{\frac{5}{7}} A(3,3) $| |
| | ^$ {Y_{1}^{(2)}} $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} A(5,4) $|$ \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)+\frac{4}{33} \sqrt{5} A(5,3) $|$ -\frac{A(3,2)}{\sqrt{21}}-\frac{5 A(5,2)}{11 \sqrt{3}} $|$ -\sqrt{\frac{3}{35}} A(1,1)+\frac{1}{3} \sqrt{\frac{2}{35}} A(3,1)+\frac{20 A(5,1)}{33 \sqrt{7}} $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ \sqrt{\frac{2}{7}} A(1,1)+\frac{A(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} A(5,1) $|$ \frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,2)-7 A(5,2)\right) $| |
| | ^$ {Y_{2}^{(2)}} $|$ -\frac{5}{11} \sqrt{\frac{2}{3}} A(5,5) $|$ \frac{1}{11} \sqrt{10} A(5,4) $|$ \frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)-\frac{5}{33} \sqrt{2} A(5,3) $|$ \frac{5}{33} A(5,2)-\frac{2 A(3,2)}{3 \sqrt{7}} $|$ -\frac{A(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} A(3,1)-\frac{5 A(5,1)}{11 \sqrt{21}} $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ \frac{1}{231} \left(33 \sqrt{21} A(1,1)-11 \sqrt{14} A(3,1)+\sqrt{35} A(5,1)\right) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ |
| | ^$ d_{z^2-x^2} $|$ 0 $|$ \frac{\sqrt{105} (99 A(1,0)+44 A(3,0)+50 A(5,0))+5 \sqrt{2} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)}{2310} $|$ 3 \sqrt{\frac{3}{70}} A(1,1)-\frac{A(3,1)}{6 \sqrt{35}}+\frac{1}{2} \sqrt{\frac{3}{7}} A(3,3)+\frac{5 A(5,1)}{6 \sqrt{14}}-\frac{5}{44} \sqrt{3} A(5,3)+\frac{5}{44} \sqrt{\frac{5}{3}} A(5,5) $|$ 0 $|$ \frac{\sqrt{14} (-33 A(1,0)+22 A(3,0)-5 A(5,0))+42 \sqrt{15} A(5,2)-42 \sqrt{5} A(5,4)}{462 \sqrt{2}} $|$ \frac{1}{924} \left(66 \sqrt{14} A(1,1)+22 \sqrt{21} A(3,1)-22 \sqrt{35} A(3,3)+17 \sqrt{210} A(5,1)+7 \sqrt{5} A(5,3)-105 A(5,5)\right) $|$ 0 $| |
| | ^$ d_{3y^2-r^2} $|$ 0 $|$ \frac{5 \sqrt{6} \left(22 \sqrt{7} A(3,2)-35 A(5,2)\right)-\sqrt{35} (99 A(1,0)+44 A(3,0)+50 A(5,0))}{2310} $|$ \frac{198 \sqrt{70} A(1,1)-154 \sqrt{105} A(3,1)-5 \left(22 \sqrt{7} A(3,3)+5 \sqrt{42} A(5,1)+35 A(5,3)-105 \sqrt{5} A(5,5)\right)}{4620} $|$ 0 $|$ \frac{\sqrt{42} (-33 A(1,0)+22 A(3,0)-5 A(5,0))-42 \sqrt{5} A(5,2)-42 \sqrt{15} A(5,4)}{462 \sqrt{2}} $|$ \frac{1}{924} \left(-66 \sqrt{42} A(1,1)-66 \sqrt{7} A(3,1)+22 \sqrt{105} A(3,3)-9 \sqrt{70} A(5,1)-49 \sqrt{15} A(5,3)-105 \sqrt{3} A(5,5)\right) $|$ 0 $| |
| | ^$ d_{\text{xy}} $|$ \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right) $|$ 0 $|$ 0 $|$ \frac{66 \sqrt{210} 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 $|$ 0 $|$ \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) $| |
| | ^$ 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 $|$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $|$ 0 $|$ 0 $|$ \frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right) $| |
| | ^$ d_{\text{xz}} $|$ 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) $|$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $|$ 0 $|$ \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) $|$ \frac{1}{462} \left(-66 \sqrt{7} A(1,0)-11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)+25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)-21 \sqrt{10} A(5,4)\right) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | |
| | ===== Table of several point groups ===== |
| | |
| | ### |
| | |
| | [[physics_chemistry:point_groups|Return to Main page on Point Groups]] |
| | |
| | ### |
| | |
| | ### |
| | |
| | ^Nonaxial groups | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> | | | | | |
| | ^C<sub>n</sub> groups | [[physics_chemistry:point_groups:c2|C]]<sub>[[physics_chemistry:point_groups:c2|2]]</sub> | [[physics_chemistry:point_groups:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> | |
| | ^D<sub>n</sub> groups | [[physics_chemistry:point_groups:d2|D]]<sub>[[physics_chemistry:point_groups:d2|2]]</sub> | [[physics_chemistry:point_groups:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub> | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub> | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub> | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub> | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub> | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub> | |
| | ^C<sub>nv</sub> groups | [[physics_chemistry:point_groups:c2v|C]]<sub>[[physics_chemistry:point_groups:c2v|2v]]</sub> | [[physics_chemistry:point_groups:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> | |
| | ^C<sub>nh</sub> groups | [[physics_chemistry:point_groups:c2h|C]]<sub>[[physics_chemistry:point_groups:c2h|2h]]</sub> | [[physics_chemistry:point_groups:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> | | | |
| | ^D<sub>nh</sub> groups | [[physics_chemistry:point_groups:d2h|D]]<sub>[[physics_chemistry:point_groups:d2h|2h]]</sub> | [[physics_chemistry:point_groups:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> | |
| | ^D<sub>nd</sub> groups | [[physics_chemistry:point_groups:d2d|D]]<sub>[[physics_chemistry:point_groups:d2d|2d]]</sub> | [[physics_chemistry:point_groups:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> | |
| | ^S<sub>n</sub> groups | [[physics_chemistry:point_groups:S2|S]]<sub>[[physics_chemistry:point_groups:S2|2]]</sub> | [[physics_chemistry:point_groups:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> | | |
| | ^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> | |
| | ^Linear groups | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | | |
| | |
| | ### |
