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| physics_chemistry:point_groups:d3d:orientation_111 [2018/03/21 18:36] – created Stefano Agrestini | physics_chemistry:point_groups:d3d:orientation_111 [2018/09/06 12:59] (current) – Maurits W. Haverkort |
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| | ~~CLOSETOC~~ |
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| ====== Orientation 111 ====== | ====== Orientation 111 ====== |
| |
| ### | ### |
| alligned paragraph text | |
| | This orientation is non-standard, but related to the orientation of the Oh pointgroup, which normally would be orrientated with the C3 axes in the 111 direction. We only show one of the options of the D3d subgroups of the Oh group with orientation XYZ. |
| ### | ### |
| |
| ===== Example ===== | ===== Symmetry Operations ===== |
| |
| ### | ### |
| description text | |
| | In the D3d Point Group, with orientation 111 there are the following symmetry operations |
| ### | ### |
| |
| ==== Input ==== | ### |
| <code Quanty Example.Quanty> | |
| -- some example code | {{:physics_chemistry:pointgroup:d3d_111.png}} |
| | |
| | ### |
| | |
| | ### |
| | |
| | ^ Operator ^ Orientation ^ |
| | ^ $\text{E}$ | $\{0,0,0\}$ , | |
| | ^ $C_3$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , | |
| | ^ $C_2$ | $\{1,-1,0\}$ , $\{0,1,-1\}$ , $\{1,0,-1\}$ , | |
| | ^ $\text{i}$ | $\{0,0,0\}$ , | |
| | ^ $S_6$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , | |
| | ^ $\sigma _d$ | $\{1,-1,0\}$ , $\{0,1,-1\}$ , $\{1,0,-1\}$ , | |
| | |
| | ### |
| | |
| | ===== Different Settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:d3d:orientation_111|Point Group D3d with orientation 111]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_zx|Point Group D3d with orientation Zx]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_zx_a|Point Group D3d with orientation Zx_A]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_zx_b|Point Group D3d with orientation Zx_B]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)|Point Group D3d with orientation Z(x-y)]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a|Point Group D3d with orientation Z(x-y)_A]] |
| | * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b|Point Group D3d with orientation Z(x-y)_B]] |
| | * [[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]] |
| | |
| | ### |
| | |
| | ===== Character Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_3 \,{\text{(2)}} $ ^ $ C_2 \,{\text{(3)}} $ ^ $ \text{i} \,{\text{(1)}} $ ^ $ S_6 \,{\text{(2)}} $ ^ $ \sigma_d \,{\text{(3)}} $ ^ |
| | ^ $ A_{1g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | |
| | ^ $ A_{2g} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | |
| | ^ $ E_g $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 2 $ | $ -1 $ | $ 0 $ | |
| | ^ $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | |
| | ^ $ A_{2u} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ 1 $ | |
| | ^ $ E_u $ | $ 2 $ | $ -1 $ | $ 0 $ | $ -2 $ | $ 1 $ | $ 0 $ | |
| | |
| | ### |
| | |
| | ===== Product Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ A_{1g} $ ^ $ A_{2g} $ ^ $ E_g $ ^ $ A_{1u} $ ^ $ A_{2u} $ ^ $ E_u $ ^ |
| | ^ $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | |
| | ^ $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | |
| | ^ $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | |
| | ^ $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | |
| | ^ $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | |
| | ^ $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | |
| | |
| | ### |
| | |
| | ===== Sub Groups with compatible settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] |
| | * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]] |
| | |
| | ### |
| | |
| | ===== Super Groups with compatible settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] |
| | |
| | ### |
| | |
| | ===== Invariant Potential expanded on renormalized spherical Harmonics ===== |
| | |
| | ### |
| | |
| | Any potential (function) can be written as a sum over spherical harmonics. |
| | $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ |
| | Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ |
| | The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the D3d Point group with orientation 111 the form of the expansion coefficients is: |
| | |
| | ### |
| | |
| | ==== Expansion ==== |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -i A(2,1) & k=2\land m=-2 \\ |
| | (-1-i) A(2,1) & k=2\land m=-1 \\ |
| | (1-i) A(2,1) & k=2\land m=1 \\ |
| | i A(2,1) & k=2\land m=2 \\ |
| | \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ |
| | (-1+i) \sqrt{7} A(4,1) & k=4\land m=-3 \\ |
| | 2 i \sqrt{2} A(4,1) & k=4\land m=-2 \\ |
| | (-1-i) A(4,1) & k=4\land m=-1 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | (1-i) A(4,1) & k=4\land m=1 \\ |
| | -2 i \sqrt{2} A(4,1) & k=4\land m=2 \\ |
| | (1+i) \sqrt{7} A(4,1) & k=4\land m=3 \\ |
| | -\frac{1}{33} i \left(8 \sqrt{22} A(6,1)-\sqrt{55} B(6,2)\right) & k=6\land m=-6 \\ |
| | -\frac{(1+i) \left(A(6,1)+2 \sqrt{10} B(6,2)\right)}{\sqrt{66}} & k=6\land m=-5 \\ |
| | -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ |
| | \left(\frac{1}{6}-\frac{i}{6}\right) \left(\sqrt{10} A(6,1)-4 B(6,2)\right) & k=6\land m=-3 \\ |
| | -i B(6,2) & k=6\land m=-2 \\ |
| | (-1-i) A(6,1) & k=6\land m=-1 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | (1-i) A(6,1) & k=6\land m=1 \\ |
| | i B(6,2) & k=6\land m=2 \\ |
| | \left(-\frac{1}{6}-\frac{i}{6}\right) \left(\sqrt{10} A(6,1)-4 B(6,2)\right) & k=6\land m=3 \\ |
| | \frac{(1-i) \left(A(6,1)+2 \sqrt{10} B(6,2)\right)}{\sqrt{66}} & k=6\land m=5 \\ |
| | \frac{1}{33} i \left(8 \sqrt{22} A(6,1)-\sqrt{55} B(6,2)\right) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(-I)*A[2, 1], k == 2 && m == -2}, {(-1 - I)*A[2, 1], k == 2 && m == -1}, {(1 - I)*A[2, 1], k == 2 && m == 1}, {I*A[2, 1], k == 2 && m == 2}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}, {(-1 + I)*Sqrt[7]*A[4, 1], k == 4 && m == -3}, {(2*I)*Sqrt[2]*A[4, 1], k == 4 && m == -2}, {(-1 - I)*A[4, 1], k == 4 && m == -1}, {A[4, 0], k == 4 && m == 0}, {(1 - I)*A[4, 1], k == 4 && m == 1}, {(-2*I)*Sqrt[2]*A[4, 1], k == 4 && m == 2}, {(1 + I)*Sqrt[7]*A[4, 1], k == 4 && m == 3}, {(-I/33)*(8*Sqrt[22]*A[6, 1] - Sqrt[55]*B[6, 2]), k == 6 && m == -6}, {((-1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6, 2]))/Sqrt[66], k == 6 && m == -5}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {(1/6 - I/6)*(Sqrt[10]*A[6, 1] - 4*B[6, 2]), k == 6 && m == -3}, {(-I)*B[6, 2], k == 6 && m == -2}, {(-1 - I)*A[6, 1], k == 6 && m == -1}, {A[6, 0], k == 6 && m == 0}, {(1 - I)*A[6, 1], k == 6 && m == 1}, {I*B[6, 2], k == 6 && m == 2}, {(-1/6 - I/6)*(Sqrt[10]*A[6, 1] - 4*B[6, 2]), k == 6 && m == 3}, {((1 - I)*(A[6, 1] + 2*Sqrt[10]*B[6, 2]))/Sqrt[66], k == 6 && m == 5}, {(I/33)*(8*Sqrt[22]*A[6, 1] - Sqrt[55]*B[6, 2]), k == 6 && m == 6}}, 0] |
| </code> | </code> |
| |
| ==== Result ==== | ### |
| <WRAP center box 100%> | |
| text produced as output | |
| </WRAP> | |
| |
| ===== Table of contents ===== | ==== Input format suitable for Quanty ==== |
| {{indexmenu>.#1}} | |
| |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {2,-1, (-1+-1*I)*(A(2,1))} , |
| | {2, 1, (1+-1*I)*(A(2,1))} , |
| | {2,-2, (-I)*(A(2,1))} , |
| | {2, 2, (I)*(A(2,1))} , |
| | {4, 0, A(4,0)} , |
| | {4,-1, (-1+-1*I)*(A(4,1))} , |
| | {4, 1, (1+-1*I)*(A(4,1))} , |
| | {4, 2, (-2*I)*((sqrt(2))*(A(4,1)))} , |
| | {4,-2, (2*I)*((sqrt(2))*(A(4,1)))} , |
| | {4,-3, (-1+1*I)*((sqrt(7))*(A(4,1)))} , |
| | {4, 3, (1+1*I)*((sqrt(7))*(A(4,1)))} , |
| | {4,-4, (sqrt(5/14))*(A(4,0))} , |
| | {4, 4, (sqrt(5/14))*(A(4,0))} , |
| | {6, 0, A(6,0)} , |
| | {6,-1, (-1+-1*I)*(A(6,1))} , |
| | {6, 1, (1+-1*I)*(A(6,1))} , |
| | {6,-2, (-I)*(B(6,2))} , |
| | {6, 2, (I)*(B(6,2))} , |
| | {6, 3, (-1/6+-1/6*I)*((sqrt(10))*(A(6,1)) + (-4)*(B(6,2)))} , |
| | {6,-3, (1/6+-1/6*I)*((sqrt(10))*(A(6,1)) + (-4)*(B(6,2)))} , |
| | {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , |
| | {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} , |
| | {6,-5, (-1+-1*I)*((1/(sqrt(66)))*(A(6,1) + (2)*((sqrt(10))*(B(6,2)))))} , |
| | {6, 5, (1+-1*I)*((1/(sqrt(66)))*(A(6,1) + (2)*((sqrt(10))*(B(6,2)))))} , |
| | {6,-6, (-1/33*I)*((8)*((sqrt(22))*(A(6,1))) + (-1)*((sqrt(55))*(B(6,2))))} , |
| | {6, 6, (1/33*I)*((8)*((sqrt(22))*(A(6,1))) + (-1)*((sqrt(55))*(B(6,2))))} } |
| | |
| | </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}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i \text{Asd}(2,1)}{\sqrt{5}} $|$ -\frac{(1-i) \text{Asd}(2,1)}{\sqrt{5}} $|$ 0 $|$ \frac{(1+i) \text{Asd}(2,1)}{\sqrt{5}} $|$ -\frac{i \text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ \left(-\frac{1}{5}-\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $|$ \frac{1}{5} i \sqrt{6} \text{App}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{3 i \text{Apf}(2,1)}{\sqrt{35}}+\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $|$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}}-(1-i) \sqrt{\frac{6}{35}} \text{Apf}(2,1) $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \frac{\left(\frac{3}{5}+\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}}-\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1) $|$ -\frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)-\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $| |
| | ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{5}+\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $|$ \text{App}(0,0) $|$ \left(\frac{1}{5}+\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $|$ i \sqrt{\frac{3}{35}} \text{Apf}(2,1)-\frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1) $|$ \left(-\frac{2}{5}+\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)-\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \left(\frac{2}{5}+\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)+\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ \frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1)-i \sqrt{\frac{3}{35}} \text{Apf}(2,1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $| |
| | ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} i \sqrt{6} \text{App}(2,1) $|$ \left(\frac{1}{5}-\frac{i}{5}\right) \sqrt{3} \text{App}(2,1) $|$ \text{App}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $|$ \frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)+\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1)-\frac{\left(\frac{3}{5}-\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}} $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ (1+i) \sqrt{\frac{6}{35}} \text{Apf}(2,1)-\frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}} $|$ -\frac{3 i \text{Apf}(2,1)}{\sqrt{35}}-\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $| |
| | ^$ {Y_{-2}^{(2)}} $|$ -\frac{i \text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$ \left(\frac{1}{21}+\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1)-\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1) $|$ \frac{2}{7} i \text{Add}(2,1)+\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $|$ \frac{5}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{-1}^{(2)}} $|$ -\frac{(1+i) \text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{21}-\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1)-\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1) $|$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $|$ \left(-\frac{1}{7}-\frac{i}{7}\right) \text{Add}(2,1)-\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \frac{1}{7} i \sqrt{6} \text{Add}(2,1)-\frac{8}{21} i \sqrt{5} \text{Add}(4,1) $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{0}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{2}{7} i \text{Add}(2,1)-\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \left(-\frac{1}{7}+\frac{i}{7}\right) \text{Add}(2,1)-\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$ \left(\frac{1}{7}+\frac{i}{7}\right) \text{Add}(2,1)+\left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \frac{2}{7} i \text{Add}(2,1)+\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{1}^{(2)}} $|$ \frac{(1-i) \text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $|$ \frac{8}{21} i \sqrt{5} \text{Add}(4,1)-\frac{1}{7} i \sqrt{6} \text{Add}(2,1) $|$ \left(\frac{1}{7}-\frac{i}{7}\right) \text{Add}(2,1)+\left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $|$ \left(\frac{1}{7}+\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1)-\left(\frac{1}{21}+\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{i \text{Asd}(2,1)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{5}{21} \text{Add}(4,0) $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{5} \text{Add}(4,1) $|$ -\frac{2}{7} i \text{Add}(2,1)-\frac{2}{7} i \sqrt{\frac{10}{3}} \text{Add}(4,1) $|$ \left(\frac{1}{7}-\frac{i}{7}\right) \sqrt{6} \text{Add}(2,1)-\left(\frac{1}{21}-\frac{i}{21}\right) \sqrt{5} \text{Add}(4,1) $|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{3 i \text{Apf}(2,1)}{\sqrt{35}}-\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \text{Aff}(2,1)+\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)-\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $|$ \frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)+\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)+\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $|$ \left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)+\left(\frac{7}{11}-\frac{7 i}{11}\right) \text{Aff}(4,1) $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ \left(-\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $|$ \frac{10}{429} i \sqrt{\frac{7}{33}} \left(8 \sqrt{22} \text{Aff}(6,1)-\sqrt{55} \text{Bff}(6,2)\right) $| |
| | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}}-(1+i) \sqrt{\frac{6}{35}} \text{Apf}(2,1) $|$ \frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1)-i \sqrt{\frac{3}{35}} \text{Apf}(2,1) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{3}+\frac{i}{3}\right) \text{Aff}(2,1)+\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)-\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ -\frac{(1+i) \text{Aff}(2,1)}{\sqrt{15}}-\left(\frac{4}{33}+\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)+\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ \frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}-\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)-\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $|$ \left(\frac{7}{33}-\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $|$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $|$ \left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $| |
| | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ \left(-\frac{2}{5}-\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)-\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ -\frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)-\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)-\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $|$ -\frac{(1-i) \text{Aff}(2,1)}{\sqrt{15}}-\left(\frac{4}{33}-\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)+\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ \left(-\frac{1}{15}-\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)-\left(\frac{25}{429}+\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $|$ \frac{2}{5} i \sqrt{\frac{2}{3}} \text{Aff}(2,1)-\frac{8}{33} i \sqrt{5} \text{Aff}(4,1)+\frac{10}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ \left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)-\left(\frac{7}{33}-\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1) $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $| |
| | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{\left(\frac{3}{5}-\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}}-\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1) $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{10}{21}} \text{Apf}(4,1)-\frac{\left(\frac{3}{5}+\frac{3 i}{5}\right) \text{Apf}(2,1)}{\sqrt{7}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)+\left(\frac{7}{11}+\frac{7 i}{11}\right) \text{Aff}(4,1) $|$ -\frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}+\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)+\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $|$ \left(-\frac{1}{15}+\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)-\left(\frac{25}{429}-\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ \left(\frac{1}{15}+\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)+\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)+\left(\frac{25}{429}+\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $|$ \frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}-\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)-\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $|$ \left(-\frac{7}{11}+\frac{7 i}{11}\right) \text{Aff}(4,1)-\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $| |
| | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} i \sqrt{\frac{3}{7}} \text{Apf}(2,1)+\frac{2}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,1) $|$ \left(\frac{2}{5}-\frac{2 i}{5}\right) \sqrt{\frac{6}{7}} \text{Apf}(2,1)+\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Apf}(4,1) $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ \left(\frac{7}{33}+\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $|$ -\frac{2}{5} i \sqrt{\frac{2}{3}} \text{Aff}(2,1)+\frac{8}{33} i \sqrt{5} \text{Aff}(4,1)-\frac{10}{143} i \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ \left(\frac{1}{15}-\frac{i}{15}\right) \sqrt{2} \text{Aff}(2,1)+\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{5}{3}} \text{Aff}(4,1)+\left(\frac{25}{429}-\frac{25 i}{429}\right) \sqrt{14} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ \frac{(1+i) \text{Aff}(2,1)}{\sqrt{15}}+\left(\frac{4}{33}+\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ \frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)+\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)+\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $| |
| | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{7} \text{Apf}(4,1) $|$ i \sqrt{\frac{3}{35}} \text{Apf}(2,1)-\frac{4}{3} i \sqrt{\frac{2}{7}} \text{Apf}(4,1) $|$ (1-i) \sqrt{\frac{6}{35}} \text{Apf}(2,1)-\frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,1)}{\sqrt{7}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $|$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $|$ \left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{\frac{7}{6}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right)-\left(\frac{7}{33}+\frac{7 i}{33}\right) \sqrt{2} \text{Aff}(4,1) $|$ -\frac{2 i \text{Aff}(2,1)}{3 \sqrt{5}}+\frac{2}{11} i \sqrt{\frac{2}{3}} \text{Aff}(4,1)+\frac{20}{429} i \sqrt{14} \text{Bff}(6,2) $|$ \frac{(1-i) \text{Aff}(2,1)}{\sqrt{15}}+\left(\frac{4}{33}-\frac{4 i}{33}\right) \sqrt{2} \text{Aff}(4,1)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{\frac{35}{3}} \text{Aff}(6,1) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \text{Aff}(2,1)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)+\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $| |
| | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{7}{3}} \text{Apf}(4,1) $|$ \frac{3 i \text{Apf}(2,1)}{\sqrt{35}}+\frac{2}{3} i \sqrt{\frac{2}{21}} \text{Apf}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{429} i \sqrt{\frac{7}{33}} \left(8 \sqrt{22} \text{Aff}(6,1)-\sqrt{55} \text{Bff}(6,2)\right) $|$ \left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \left(\text{Aff}(6,1)+2 \sqrt{10} \text{Bff}(6,2)\right) $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ \left(-\frac{7}{11}-\frac{7 i}{11}\right) \text{Aff}(4,1)-\left(\frac{5}{429}+\frac{5 i}{429}\right) \sqrt{\frac{7}{3}} \left(\sqrt{10} \text{Aff}(6,1)-4 \text{Bff}(6,2)\right) $|$ -\frac{1}{3} i \sqrt{\frac{2}{5}} \text{Aff}(2,1)-\frac{4}{11} i \sqrt{3} \text{Aff}(4,1)-\frac{10}{429} i \sqrt{7} \text{Bff}(6,2) $|$ \left(\frac{1}{3}-\frac{i}{3}\right) \text{Aff}(2,1)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{\frac{10}{3}} \text{Aff}(4,1)+\left(\frac{5}{429}-\frac{5 i}{429}\right) \sqrt{7} \text{Aff}(6,1) $|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| |
| | |
| | |
| | ### |
| | |
| | ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== |
| | |
| | ### |
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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_{x+y+z} $|$\color{darkred}{ 0 }$|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1-i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_{x-y} $|$\color{darkred}{ 0 }$|$ \frac{1}{2}-\frac{i}{2} $|$ 0 $|$ -\frac{1}{2}-\frac{i}{2} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_{3z-r} $|$\color{darkred}{ 0 }$|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \sqrt{\frac{2}{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ \frac{1+i}{\sqrt{6}} $|$ 0 $|$ -\frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}-\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|$ 0 $|$ \frac{1}{2}+\frac{i}{2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x^3+y^3+z^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{3}} $|$ \frac{1}{4}-\frac{i}{4} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $| |
| | ^$ f_{x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $| |
| | ^$ f_{2z^3-x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \sqrt{\frac{2}{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $| |
| | ^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{6}} $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{4}-\frac{i}{4} $| |
| | ^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \frac{1}{\sqrt{3}} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $| |
| | ^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $| |
| | |
| | |
| | ### |
| | |
| | ==== One particle coupling on a basis of symmetry adapted functions ==== |
| | |
| | ### |
| | |
| | After rotation we find |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{s} $ ^ $ p_{x+y+z} $ ^ $ p_{x-y} $ ^ $ p_{3z-r} $ ^ $ d_{\text{yz}+\text{xz}+\text{xy}} $ ^ $ d_{\text{yz}-\text{xz}} $ ^ $ d_{2\text{xy}-\text{xz}-\text{yz}} $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ f_{\text{xyz}} $ ^ $ f_{x^3+y^3+z^3} $ ^ $ f_{x^3-y^3} $ ^ $ f_{2z^3-x^3-y^3} $ ^ $ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $ ^ $ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $ ^ $ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $ ^ |
| | ^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{6}{5}} \text{Asd}(2,1) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ p_{x+y+z} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{2}{5} \sqrt{6} \text{App}(2,1) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{8 \text{Apf}(4,1)}{\sqrt{21}}-3 \sqrt{\frac{2}{35}} \text{Apf}(2,1) $|$ \frac{6}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}-\frac{4}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_{x-y} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)+\frac{1}{5} \sqrt{6} \text{App}(2,1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ 0 $|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $|$ 0 $| |
| | ^$ p_{3z-r} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)+\frac{1}{5} \sqrt{6} \text{App}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ 0 $|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $| |
| | ^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ -\sqrt{\frac{6}{5}} \text{Asd}(2,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \sqrt{6} \text{Add}(2,1)-\frac{4}{21} \text{Add}(4,0)+\frac{16}{21} \sqrt{5} \text{Add}(4,1) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}-\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,1)-\frac{4}{21} \text{Add}(4,0)-\frac{8}{21} \sqrt{5} \text{Add}(4,1) $|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,1)-\frac{4}{21} \text{Add}(4,0)-\frac{8}{21} \sqrt{5} \text{Add}(4,1) $|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{2}{7} \sqrt{3} \text{Add}(2,1)+\frac{2}{7} \sqrt{10} \text{Add}(4,1) $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ \frac{8 \text{Apf}(4,1)}{\sqrt{21}}-3 \sqrt{\frac{2}{35}} \text{Apf}(2,1) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{4}{11} \text{Aff}(4,0)+\frac{80}{143} \text{Aff}(6,0) $|$ 2 \sqrt{\frac{2}{15}} \text{Aff}(2,1)-\frac{4}{11} \text{Aff}(4,1)-\frac{40}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x^3+y^3+z^3} $|$\color{darkred}{ 0 }$|$ \frac{6}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}-\frac{4}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 2 \sqrt{\frac{2}{15}} \text{Aff}(2,1)-\frac{4}{11} \text{Aff}(4,1)-\frac{40}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $|$ \text{Aff}(0,0)+\frac{1}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,1)+\frac{2}{11} \text{Aff}(4,0)+\frac{8}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)+\frac{100}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{20}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $|$ 0 $| |
| | ^$ f_{2z^3-x^3-y^3} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,1)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}+\frac{2}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{5 \sqrt{6}}+\frac{2}{11} \text{Aff}(4,0)-\frac{4}{11} \sqrt{5} \text{Aff}(4,1)+\frac{100}{429} \text{Aff}(6,0)-\frac{50}{429} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{10}{429} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $| |
| | ^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\sqrt{\frac{2}{3}} \text{Aff}(2,1)-\frac{2}{33} \text{Aff}(4,0)+\frac{8}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)-\frac{20}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)+\frac{20}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $| |
| | ^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$\color{darkred}{ 0 }$|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $|$ 0 $| |
| | ^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -3 \sqrt{\frac{2}{35}} \text{Apf}(2,1)-2 \sqrt{\frac{3}{7}} \text{Apf}(4,1) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,1)}{\sqrt{30}}+\frac{8}{11} \text{Aff}(4,1)-\frac{10}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{7}{3}} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{\text{Aff}(2,1)}{\sqrt{6}}-\frac{2}{33} \text{Aff}(4,0)-\frac{4}{33} \sqrt{5} \text{Aff}(4,1)-\frac{60}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,1)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Bff}(6,2) $| |
| | |
| | |
| | ### |
| | |
| | ===== Coupling for a single shell ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$. |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| | |
| | ==== Potential for s orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \text{Ea1g} & k=0\land m=0 \\ |
| | 0 & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{0, 0, Ea1g} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \text{Ea1g} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{s} $ ^ |
| | ^$ \text{s} $|$ \text{Ea1g} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ \text{s} $|$ 1 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d3d_111_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** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\ |
| | \frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=-2 \\ |
| | \frac{\left(\frac{5}{3}+\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=-1 \\ |
| | -\frac{\left(\frac{5}{3}-\frac{5 i}{3}\right) (\text{Ea2u}-\text{Eeu})}{\sqrt{6}} & k=2\land m=1 \\ |
| | -\frac{5 i (\text{Ea2u}-\text{Eeu})}{3 \sqrt{6}} & k=2\land m=2 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(((5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == -2}, {((5/3 + (5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == -1}, {((-5/3 + (5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == 1}, {(((-5*I)/3)*(Ea2u - Eeu))/Sqrt[6], k == 2 && m == 2}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , |
| | {2, 1, (-5/3+5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , |
| | {2,-1, (5/3+5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , |
| | {2, 2, (-5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} , |
| | {2,-2, (5/3*I)*((1/(sqrt(6)))*(Ea2u + (-1)*(Eeu)))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $|$ \frac{1}{3} \sqrt[4]{-1} (\text{Ea2u}-\text{Eeu}) $|$ -\frac{1}{3} i (\text{Ea2u}-\text{Eeu}) $| |
| | ^$ {Y_{0}^{(1)}} $|$ \frac{1}{3} (-1)^{3/4} (\text{Eeu}-\text{Ea2u}) $|$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $|$ \frac{1}{3} \sqrt[4]{-1} (\text{Eeu}-\text{Ea2u}) $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{1}{3} i (\text{Ea2u}-\text{Eeu}) $|$ \frac{1}{3} (-1)^{3/4} (\text{Ea2u}-\text{Eeu}) $|$ \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_{x+y+z} $ ^ $ p_{x-y} $ ^ $ p_{3z-r} $ ^ |
| | ^$ p_{x+y+z} $|$ \text{Ea2u} $|$ 0 $|$ 0 $| |
| | ^$ p_{x-y} $|$ 0 $|$ \text{Eeu} $|$ 0 $| |
| | ^$ p_{3z-r} $|$ 0 $|$ 0 $|$ \text{Eeu} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ p_{x+y+z} $|$ \frac{1+i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1-i}{\sqrt{6}} $| |
| | ^$ p_{x-y} $|$ \frac{1}{2}-\frac{i}{2} $|$ 0 $|$ -\frac{1}{2}-\frac{i}{2} $| |
| | ^$ p_{3z-r} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ \sqrt{\frac{2}{3}} $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_1_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta )}{2 \sqrt{\pi }}$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{x+y+z}{2 \sqrt{\pi }}$$ | ::: | |
| | ^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_1_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi ))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{2 \pi }} (x-y)$$ | ::: | |
| | ^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_1_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta )}{2 \sqrt{2 \pi }}$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{x+y-2 z}{2 \sqrt{2 \pi }}$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{5} (\text{Ea1g}+2 (\text{Eeg$\pi $}+\text{Eeg$\sigma $})) & k=0\land m=0 \\ |
| | \frac{1}{6} i \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg$\pi $}-4 \sqrt{3} \text{Meg}\right) & k=2\land m=-2 \\ |
| | \left(\frac{1}{6}+\frac{i}{6}\right) \left(\sqrt{6} \text{Ea1g}-\sqrt{6} \text{Eeg$\pi $}-4 \sqrt{3} \text{Meg}\right) & k=2\land m=-1 \\ |
| | \left(\frac{1}{6}-\frac{i}{6}\right) \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg$\pi $}+4 \sqrt{3} \text{Meg}\right) & k=2\land m=1 \\ |
| | \frac{1}{6} i \left(-\sqrt{6} \text{Ea1g}+\sqrt{6} \text{Eeg$\pi $}+4 \sqrt{3} \text{Meg}\right) & k=2\land m=2 \\ |
| | -\frac{1}{2} \sqrt{\frac{7}{10}} (\text{Ea1g}+2 \text{Eeg$\pi $}-3 \text{Eeg$\sigma $}) & k=4\land (m=-4\lor m=4) \\ |
| | \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) & k=4\land m=-3 \\ |
| | \frac{i \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=-2 \\ |
| | -\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=-1 \\ |
| | -\frac{7}{10} (\text{Ea1g}+2 \text{Eeg$\pi $}-3 \text{Eeg$\sigma $}) & k=4\land m=0 \\ |
| | \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{5}} & k=4\land m=1 \\ |
| | -\frac{i \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right)}{\sqrt{10}} & k=4\land m=2 \\ |
| | \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{7}{5}} \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) & k=4\land m=3 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Ea1g + 2*(Eeg\[Pi] + Eeg\[Sigma]))/5, k == 0 && m == 0}, {(I/6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg), k == 2 && m == -2}, {(1/6 + I/6)*(Sqrt[6]*Ea1g - Sqrt[6]*Eeg\[Pi] - 4*Sqrt[3]*Meg), k == 2 && m == -1}, {(1/6 - I/6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg), k == 2 && m == 1}, {(I/6)*(-(Sqrt[6]*Ea1g) + Sqrt[6]*Eeg\[Pi] + 4*Sqrt[3]*Meg), k == 2 && m == 2}, {-(Sqrt[7/10]*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/2, k == 4 && (m == -4 || m == 4)}, {(-1/4 + I/4)*Sqrt[7/5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg), k == 4 && m == -3}, {(I*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[10], k == 4 && m == -2}, {((-1/4 - I/4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[5], k == 4 && m == -1}, {(-7*(Ea1g + 2*Eeg\[Pi] - 3*Eeg\[Sigma]))/10, k == 4 && m == 0}, {((1/4 - I/4)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[5], k == 4 && m == 1}, {((-I)*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg))/Sqrt[10], k == 4 && m == 2}, {(1/4 + I/4)*Sqrt[7/5]*(2*Ea1g - 2*Eeg\[Pi] + 3*Sqrt[2]*Meg), k == 4 && m == 3}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg\[Pi] + Eeg\[Sigma]))} , |
| | {2, 1, (1/6+-1/6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , |
| | {2,-1, (1/6+1/6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , |
| | {2, 2, (1/6*I)*((-1)*((sqrt(6))*(Ea1g)) + (sqrt(6))*(Eeg\[Pi]) + (4)*((sqrt(3))*(Meg)))} , |
| | {2,-2, (1/6*I)*((sqrt(6))*(Ea1g) + (-1)*((sqrt(6))*(Eeg\[Pi])) + (-4)*((sqrt(3))*(Meg)))} , |
| | {4, 0, (-7/10)*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma]))} , |
| | {4,-1, (-1/4+-1/4*I)*((1/(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , |
| | {4, 1, (1/4+-1/4*I)*((1/(sqrt(5)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , |
| | {4, 2, (-I)*((1/(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , |
| | {4,-2, (I)*((1/(sqrt(10)))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , |
| | {4,-3, (-1/4+1/4*I)*((sqrt(7/5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , |
| | {4, 3, (1/4+1/4*I)*((sqrt(7/5))*((2)*(Ea1g) + (-2)*(Eeg\[Pi]) + (3)*((sqrt(2))*(Meg))))} , |
| | {4,-4, (-1/2)*((sqrt(7/10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} , |
| | {4, 4, (-1/2)*((sqrt(7/10))*(Ea1g + (2)*(Eeg\[Pi]) + (-3)*(Eeg\[Sigma])))} } |
| | |
| | </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}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) $|$ \frac{i \text{Meg}}{\sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $| |
| | ^$ {Y_{-1}^{(2)}} $|$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) $|$ \text{Meg} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right] $|$ -\frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) $|$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $| |
| | ^$ {Y_{0}^{(2)}} $|$ -\frac{i \text{Meg}}{\sqrt{3}} $|$ \frac{(-1)^{3/4} \text{Meg}}{\sqrt{6}} $|$ \text{Eeg$\sigma $} $|$ \frac{\sqrt[4]{-1} \text{Meg}}{\sqrt{6}} $|$ \frac{i \text{Meg}}{\sqrt{3}} $| |
| | ^$ {Y_{1}^{(2)}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(2 \text{Ea1g}-2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{3} i (\text{Ea1g}-\text{Eeg$\pi $}) $|$ \text{Meg} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right] $|$ \frac{1}{3} (\text{Ea1g}+2 \text{Eeg$\pi $}) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{1}{6} (-\text{Ea1g}-2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}-3 \sqrt{2} \text{Meg}\right) $|$ -\frac{i \text{Meg}}{\sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(-2 \text{Ea1g}+2 \text{Eeg$\pi $}+3 \sqrt{2} \text{Meg}\right) $|$ \frac{1}{6} (\text{Ea1g}+2 \text{Eeg$\pi $}+3 \text{Eeg$\sigma $}) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{\text{yz}+\text{xz}+\text{xy}} $ ^ $ d_{\text{yz}-\text{xz}} $ ^ $ d_{2\text{xy}-\text{xz}-\text{yz}} $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ |
| | ^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ \text{Ea1g} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Meg} \left(\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(1+i)\right)}{\sqrt{2}} $| |
| | ^$ d_{\text{yz}-\text{xz}} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ \text{Meg} $|$ \left(-\frac{1}{2}-\frac{i}{2}\right) \text{Meg} \left(\text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]\right) $| |
| | ^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ 0 $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \text{Meg} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(2-2 i)\right) $| |
| | ^$ d_{x^2-y^2} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\sigma $} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ d_{\text{yz}+\text{xz}+\text{xy}} $|$ \frac{i}{\sqrt{6}} $|$ \frac{1+i}{\sqrt{6}} $|$ 0 $|$ -\frac{1-i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{6}} $| |
| | ^$ d_{\text{yz}-\text{xz}} $|$ 0 $|$ -\frac{1}{2}+\frac{i}{2} $|$ 0 $|$ \frac{1}{2}+\frac{i}{2} $|$ 0 $| |
| | ^$ d_{2\text{xy}-\text{xz}-\text{yz}} $|$ \frac{i}{\sqrt{3}} $|$ -\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}} $|$ 0 $|$ \frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}} $|$ -\frac{i}{\sqrt{3}} $| |
| | ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_2_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$$ | ::: | |
| | ^ ^$$\text{Eeg$\pi $}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_2_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{2 \pi }} \sin (2 \theta ) (\sin (\phi )-\cos (\phi ))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{2 \pi }} z (y-x)$$ | ::: | |
| | ^ ^$$\text{Eeg$\pi $}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_2_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$$ | ::: | |
| | ^ ^$$\text{Eeg$\sigma $}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_2_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | |
| | ^ ^$$\text{Eeg$\sigma $}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_2_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for f orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 (\text{Eeu1}+\text{Eeu2})) & k=0\land m=0 \\ |
| | -\frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=-2 \\ |
| | -\frac{\left(\frac{5}{28}+\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=-1 \\ |
| | \frac{\left(\frac{5}{28}-\frac{5 i}{28}\right) \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} & k=2\land m=1 \\ |
| | \frac{5 i \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+2 \sqrt{5} (2 \text{Ma2u}-\text{Meu})\right)}{28 \sqrt{6}} & k=2\land m=2 \\ |
| | -\frac{1}{4} \sqrt{\frac{5}{14}} (\text{Ea1u}+6 \text{Ea2u1}-3 \text{Ea2u2}-6 \text{Eeu1}+2 \text{Eeu2}) & k=4\land (m=-4\lor m=4) \\ |
| | -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=-3 \\ |
| | \frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=-2 \\ |
| | \left(-\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=-1 \\ |
| | \frac{1}{4} (-\text{Ea1u}-6 \text{Ea2u1}+3 \text{Ea2u2}+6 \text{Eeu1}-2 \text{Eeu2}) & k=4\land m=0 \\ |
| | \left(\frac{1}{28}-\frac{i}{28}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right) & k=4\land m=1 \\ |
| | -\frac{i \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{7 \sqrt{2}} & k=4\land m=2 \\ |
| | \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\sqrt{5} \text{Ea1u}+3 \sqrt{5} \text{Ea2u2}-3 \sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+12 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=3 \\ |
| | \frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=-6 \\ |
| | \left(-\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=-5 \\ |
| | \frac{13 (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ |
| | -\frac{\left(\frac{13}{40}-\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=-3 \\ |
| | -\frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=-2 \\ |
| | \frac{\left(\frac{13}{20}+\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=-1 \\ |
| | -\frac{13}{280} (9 \text{Ea1u}-12 \text{Ea2u1}-5 (\text{Ea2u2}+2 \text{Eeu1})+18 \text{Eeu2}) & k=6\land m=0 \\ |
| | -\frac{\left(\frac{13}{20}-\frac{13 i}{20}\right) \left(2 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-2 \text{Eeu2}+\sqrt{5} (\text{Ma2u}+7 \text{Meu})\right)}{\sqrt{42}} & k=6\land m=1 \\ |
| | \frac{13 i \left(7 \sqrt{5} \text{Ea1u}-\sqrt{5} \text{Ea2u2}+\sqrt{5} \text{Eeu1}-7 \sqrt{5} \text{Eeu2}-32 \text{Ma2u}-26 \text{Meu}\right)}{80 \sqrt{21}} & k=6\land m=2 \\ |
| | \frac{\left(\frac{13}{40}+\frac{13 i}{40}\right) \left(3 \sqrt{5} \text{Ea1u}-2 \sqrt{5} \text{Ea2u2}+2 \sqrt{5} \text{Eeu1}-3 \sqrt{5} \text{Eeu2}-9 \text{Ma2u}+3 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=3 \\ |
| | \left(\frac{13}{40}-\frac{13 i}{40}\right) \sqrt{\frac{11}{7}} \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right) & k=6\land m=5 \\ |
| | -\frac{13}{80} i \sqrt{\frac{11}{21}} \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*(Eeu1 + Eeu2))/7, k == 0 && m == 0}, {(((-5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == -2}, {((-5/28 - (5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == -1}, {((5/28 - (5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == 1}, {(((5*I)/28)*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 + 2*Sqrt[5]*(2*Ma2u - Meu)))/Sqrt[6], k == 2 && m == 2}, {-(Sqrt[5/14]*(Ea1u + 6*Ea2u1 - 3*Ea2u2 - 6*Eeu1 + 2*Eeu2))/4, k == 4 && (m == -4 || m == 4)}, {((-1/4 + I/4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[7], k == 4 && m == -3}, {((I/7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[2], k == 4 && m == -2}, {(-1/28 - I/28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == -1}, {(-Ea1u - 6*Ea2u1 + 3*Ea2u2 + 6*Eeu1 - 2*Eeu2)/4, k == 4 && m == 0}, {(1/28 - I/28)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu), k == 4 && m == 1}, {((-I/7)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[2], k == 4 && m == 2}, {((1/4 + I/4)*(Sqrt[5]*Ea1u + 3*Sqrt[5]*Ea2u2 - 3*Sqrt[5]*Eeu1 - Sqrt[5]*Eeu2 - 3*Ma2u + 12*Meu))/Sqrt[7], k == 4 && m == 3}, {((13*I)/80)*Sqrt[11/21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu), k == 6 && m == -6}, {(-13/40 - (13*I)/40)*Sqrt[11/7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == -5}, {(13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {((-13/40 + (13*I)/40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/Sqrt[21], k == 6 && m == -3}, {(((-13*I)/80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/Sqrt[21], k == 6 && m == -2}, {((13/20 + (13*I)/20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/Sqrt[42], k == 6 && m == -1}, {(-13*(9*Ea1u - 12*Ea2u1 - 5*(Ea2u2 + 2*Eeu1) + 18*Eeu2))/280, k == 6 && m == 0}, {((-13/20 + (13*I)/20)*(2*Ea1u - 5*Ea2u2 + 5*Eeu1 - 2*Eeu2 + Sqrt[5]*(Ma2u + 7*Meu)))/Sqrt[42], k == 6 && m == 1}, {(((13*I)/80)*(7*Sqrt[5]*Ea1u - Sqrt[5]*Ea2u2 + Sqrt[5]*Eeu1 - 7*Sqrt[5]*Eeu2 - 32*Ma2u - 26*Meu))/Sqrt[21], k == 6 && m == 2}, {((13/40 + (13*I)/40)*(3*Sqrt[5]*Ea1u - 2*Sqrt[5]*Ea2u2 + 2*Sqrt[5]*Eeu1 - 3*Sqrt[5]*Eeu2 - 9*Ma2u + 3*Meu))/Sqrt[21], k == 6 && m == 3}, {(13/40 - (13*I)/40)*Sqrt[11/7]*(Ea1u - Eeu2 - Sqrt[5]*(Ma2u + Meu)), k == 6 && m == 5}, {((-13*I)/80)*Sqrt[11/21]*(3*Ea1u - 5*Ea2u2 + 5*Eeu1 - 3*Eeu2 + 6*Sqrt[5]*Meu), k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1 + Eeu2))} , |
| | {2,-1, (-5/28+-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , |
| | {2, 1, (5/28+-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , |
| | {2,-2, (-5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , |
| | {2, 2, (5/28*I)*((1/(sqrt(6)))*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (2)*((sqrt(5))*((2)*(Ma2u) + (-1)*(Meu)))))} , |
| | {4, 0, (1/4)*((-1)*(Ea1u) + (-6)*(Ea2u1) + (3)*(Ea2u2) + (6)*(Eeu1) + (-2)*(Eeu2))} , |
| | {4,-1, (-1/28+-1/28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , |
| | {4, 1, (1/28+-1/28*I)*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu))} , |
| | {4, 2, (-1/7*I)*((1/(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , |
| | {4,-2, (1/7*I)*((1/(sqrt(2)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , |
| | {4,-3, (-1/4+1/4*I)*((1/(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , |
| | {4, 3, (1/4+1/4*I)*((1/(sqrt(7)))*((sqrt(5))*(Ea1u) + (3)*((sqrt(5))*(Ea2u2)) + (-3)*((sqrt(5))*(Eeu1)) + (-1)*((sqrt(5))*(Eeu2)) + (-3)*(Ma2u) + (12)*(Meu)))} , |
| | {4,-4, (-1/4)*((sqrt(5/14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , |
| | {4, 4, (-1/4)*((sqrt(5/14))*(Ea1u + (6)*(Ea2u1) + (-3)*(Ea2u2) + (-6)*(Eeu1) + (2)*(Eeu2)))} , |
| | {6, 0, (-13/280)*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2))} , |
| | {6, 1, (-13/20+13/20*I)*((1/(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , |
| | {6,-1, (13/20+13/20*I)*((1/(sqrt(42)))*((2)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-2)*(Eeu2) + (sqrt(5))*(Ma2u + (7)*(Meu))))} , |
| | {6,-2, (-13/80*I)*((1/(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , |
| | {6, 2, (13/80*I)*((1/(sqrt(21)))*((7)*((sqrt(5))*(Ea1u)) + (-1)*((sqrt(5))*(Ea2u2)) + (sqrt(5))*(Eeu1) + (-7)*((sqrt(5))*(Eeu2)) + (-32)*(Ma2u) + (-26)*(Meu)))} , |
| | {6,-3, (-13/40+13/40*I)*((1/(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , |
| | {6, 3, (13/40+13/40*I)*((1/(sqrt(21)))*((3)*((sqrt(5))*(Ea1u)) + (-2)*((sqrt(5))*(Ea2u2)) + (2)*((sqrt(5))*(Eeu1)) + (-3)*((sqrt(5))*(Eeu2)) + (-9)*(Ma2u) + (3)*(Meu)))} , |
| | {6,-4, (13/40)*((1/(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , |
| | {6, 4, (13/40)*((1/(sqrt(14)))*((9)*(Ea1u) + (-12)*(Ea2u1) + (-5)*(Ea2u2 + (2)*(Eeu1)) + (18)*(Eeu2)))} , |
| | {6,-5, (-13/40+-13/40*I)*((sqrt(11/7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , |
| | {6, 5, (13/40+-13/40*I)*((sqrt(11/7))*(Ea1u + (-1)*(Eeu2) + (-1)*((sqrt(5))*(Ma2u + Meu))))} , |
| | {6, 6, (-13/80*I)*((sqrt(11/21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} , |
| | {6,-6, (13/80*I)*((sqrt(11/21))*((3)*(Ea1u) + (-5)*(Ea2u2) + (5)*(Eeu1) + (-3)*(Eeu2) + (6)*((sqrt(5))*(Meu))))} } |
| | |
| | </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}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Meu}-\text{Ma2u})\right)}{\sqrt{6}} $|$ \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|$ \left(\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|$ -\frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $| |
| | ^$ {Y_{-2}^{(3)}} $|$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) $|$ -\frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{i \text{Ma2u}}{\sqrt{6}} $|$ \frac{1}{12} (-1)^{3/4} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) $|$ \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}-\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $| |
| | ^$ {Y_{-1}^{(3)}} $|$ -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|$ \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{1}{24} i \left(5 \text{Ea1u}-3 \text{Ea2u2}+3 \text{Eeu1}-5 \text{Eeu2}-6 \sqrt{5} \text{Meu}\right) $|$ \frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $| |
| | ^$ {Y_{0}^{(3)}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|$ -\frac{i \text{Ma2u}}{\sqrt{6}} $|$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{1}{3} (\text{Ea2u2}+2 \text{Eeu1}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea2u2}-\text{Eeu1}-\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{i \text{Ma2u}}{\sqrt{6}} $|$ \left(-\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $| |
| | ^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|$ \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}-3 \text{Ma2u}+3 \text{Meu}\right) $|$ \frac{1}{24} i \left(-5 \text{Ea1u}+3 \text{Ea2u2}-3 \text{Eeu1}+5 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $|$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea2u2}+\text{Eeu1}+\sqrt{5} \text{Meu}\right)}{\sqrt{3}} $|$ \frac{1}{24} (5 \text{Ea1u}+3 \text{Ea2u2}+6 \text{Eeu1}+10 \text{Eeu2}) $|$ \frac{1}{12} \sqrt[4]{-1} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $| |
| | ^$ {Y_{2}^{(3)}} $|$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{6} (\text{Ea1u}-3 \text{Ea2u1}+2 \text{Eeu2}) $|$ \frac{1}{12} \sqrt[4]{-1} \left(-\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Eeu2}+3 \text{Ma2u}-3 \text{Meu}\right) $|$ -\frac{i \text{Ma2u}}{\sqrt{6}} $|$ -\frac{1}{12} (-1)^{3/4} \left(\sqrt{5} \text{Ea1u}-\sqrt{5} \text{Eeu2}+3 (\text{Ma2u}+\text{Meu})\right) $|$ \frac{1}{6} (\text{Ea1u}+3 \text{Ea2u1}+2 \text{Eeu2}) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $| |
| | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{24} i \left(3 \text{Ea1u}-5 \text{Ea2u2}+5 \text{Eeu1}-3 \text{Eeu2}+6 \sqrt{5} \text{Meu}\right) $|$ -\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(-\text{Ea1u}+\text{Eeu2}+\sqrt{5} (\text{Ma2u}+\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{8} \sqrt{\frac{5}{3}} (-\text{Ea1u}+\text{Ea2u2}+2 \text{Eeu1}-2 \text{Eeu2}) $|$ \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+3 \text{Meu}\right) $|$ -\frac{i \left(\sqrt{5} \text{Ea1u}+\sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+2 \text{Meu}\right)}{8 \sqrt{3}} $|$ \frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(\text{Ea1u}-\text{Eeu2}+\sqrt{5} (\text{Ma2u}-\text{Meu})\right)}{\sqrt{6}} $|$ \frac{1}{24} (3 \text{Ea1u}+5 \text{Ea2u2}+10 \text{Eeu1}+6 \text{Eeu2}) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x^3+y^3+z^3} $ ^ $ f_{x^3-y^3} $ ^ $ f_{2z^3-x^3-y^3} $ ^ $ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $ ^ $ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $ ^ $ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $ ^ |
| | ^$ f_{\text{xyz}} $|$ \text{Ea2u1} $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x^3+y^3+z^3} $|$ \text{Ma2u} $|$ \text{Ea2u2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x^3-y^3} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $| |
| | ^$ f_{2z^3-x^3-y^3} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $| |
| | ^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1u} $|$ 0 $|$ 0 $| |
| | ^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $| |
| | ^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x^3+y^3+z^3} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{3}} $|$ \frac{1}{4}-\frac{i}{4} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} $| |
| | ^$ f_{x^3-y^3} $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $| |
| | ^$ f_{2z^3-x^3-y^3} $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \sqrt{\frac{2}{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{6}} $| |
| | ^$ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $|$ -\frac{1}{4}-\frac{i}{4} $|$ \frac{1}{\sqrt{6}} $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{4}-\frac{i}{4} $| |
| | ^$ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $|$ \frac{\frac{1}{4}+\frac{i}{4}}{\sqrt{2}} $|$ \frac{1}{\sqrt{3}} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{6}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{\frac{1}{4}-\frac{i}{4}}{\sqrt{2}} $| |
| | ^$ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $|$ \left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $|$ 0 $|$ \left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{2}} $|$ 0 $|$ \left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{3}{2}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ea2u1}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | |
| | ^ ^$$\text{Ea2u2}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{32}+\frac{i}{32}\right) \sqrt{\frac{7}{3 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{3 \pi }} \left(5 x^3-15 x^2 y-3 x \left(5 y^2+5 z^2-1\right)+5 y^3+y \left(3-15 z^2\right)+4 z \left(5 z^2-3\right)\right)$$ | ::: | |
| | ^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(5 \sin ^2(\theta ) \sin (2 \phi )-5 \cos (2 \theta )-1\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{2 \pi }} (x-y) \left(5 x^2+20 x y+5 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(-\frac{1}{32}-\frac{i}{32}\right) \sqrt{\frac{7}{6 \pi }} e^{-3 i \phi } \left(5 e^{6 i \phi } \sin ^3(\theta )-(4-4 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)-3 e^{2 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)+3 i e^{4 i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)-5 i \sin ^3(\theta )\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{6 \pi }} \left(-5 x^3+15 x^2 y+3 x \left(5 y^2+5 z^2-1\right)-5 y^3+3 y \left(5 z^2-1\right)+8 z \left(5 z^2-3\right)\right)$$ | ::: | |
| | ^ ^$$\text{Ea1u}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(-\sin ^2(\theta ) \sin (2 \phi )+\sin (2 \theta ) (\sin (\phi )+\cos (\phi ))-2 \cos ^2(\theta )\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} (x-y) \left(x^2+4 x (y-z)+y^2-4 y z+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_6.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )+2 \sin (2 \theta ) (\sin (\phi )+\cos (\phi ))+2 \cos ^2(\theta )\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{2 \pi }} (x-y) \left(x^2+4 x (y+2 z)+y^2+8 y z+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d3d_111_orb_3_7.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{105}{2 \pi }} \sin (\theta ) (\sin (\phi )+\cos (\phi )) \left(\sin ^2(\theta ) \sin (2 \phi )-2 \cos ^2(\theta )\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{2 \pi }} (x+y) \left(x^2-4 x y+y^2+5 z^2-1\right)$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ===== Coupling between two shells ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| | |
| | ==== Potential for s-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k=0\land m=0 \\ |
| | i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-2 \\ |
| | (1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=-1 \\ |
| | (-1+i) \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=1 \\ |
| | -i \sqrt{\frac{5}{6}} \text{Ma1g} & k=2\land m=2 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {I*Sqrt[5/6]*Ma1g, k == 2 && m == -2}, {(1 + I)*Sqrt[5/6]*Ma1g, k == 2 && m == -1}, {(-1 + I)*Sqrt[5/6]*Ma1g, k == 2 && m == 1}, {(-I)*Sqrt[5/6]*Ma1g, k == 2 && m == 2}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{2, 1, (-1+1*I)*((sqrt(5/6))*(Ma1g))} , |
| | {2,-1, (1+1*I)*((sqrt(5/6))*(Ma1g))} , |
| | {2, 2, (-I)*((sqrt(5/6))*(Ma1g))} , |
| | {2,-2, (I)*((sqrt(5/6))*(Ma1g))} } |
| | |
| | </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{i \text{Ma1g}}{\sqrt{6}} $|$ \frac{(1-i) \text{Ma1g}}{\sqrt{6}} $|$ 0 $|$ -\frac{(1+i) \text{Ma1g}}{\sqrt{6}} $|$ \frac{i \text{Ma1g}}{\sqrt{6}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{\text{yz}+\text{xz}+\text{xy}} $ ^ $ d_{\text{yz}-\text{xz}} $ ^ $ d_{2\text{xy}-\text{xz}-\text{yz}} $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ |
| | ^$ \text{s} $|$ \text{Ma1g} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 2\land k\neq 4)\lor (k\neq 4\land m\neq -2\land m\neq -1\land m\neq 1\land m\neq 2)\lor (m\neq -4\land m\neq -3\land m\neq -2\land m\neq -1\land m\neq 0\land m\neq 1\land m\neq 2\land m\neq 3\land m\neq 4) \\ |
| | \frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-2 \\ |
| | \left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=-1 \\ |
| | \left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=1 \\ |
| | -\frac{1}{3} i \sqrt{\frac{5}{14}} (3 \text{Ma2u1}+4 \text{Meu2}) & k=2\land m=2 \\ |
| | -\frac{1}{4} \sqrt{\frac{3}{10}} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land (m=-4\lor m=4) \\ |
| | \left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-3 \\ |
| | i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-2 \\ |
| | \left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=-1 \\ |
| | -\frac{1}{20} \sqrt{21} \left(2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}\right) & k=4\land m=0 \\ |
| | \left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{3}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=1 \\ |
| | -i \sqrt{\frac{6}{7}} (\text{Ma2u1}-\text{Meu2}) & k=4\land m=2 \\ |
| | \left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{3} (\text{Ma2u1}-\text{Meu2}) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || (k != 4 && m != -2 && m != -1 && m != 1 && m != 2) || (m != -4 && m != -3 && m != -2 && m != -1 && m != 0 && m != 1 && m != 2 && m != 3 && m != 4)}, {(I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -2}, {(1/3 + I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == -1}, {(-1/3 + I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 1}, {(-I/3)*Sqrt[5/14]*(3*Ma2u1 + 4*Meu2), k == 2 && m == 2}, {-(Sqrt[3/10]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/4, k == 4 && (m == -4 || m == 4)}, {(-1/2 + I/2)*Sqrt[3]*(Ma2u1 - Meu2), k == 4 && m == -3}, {I*Sqrt[6/7]*(Ma2u1 - Meu2), k == 4 && m == -2}, {(-1/2 - I/2)*Sqrt[3/7]*(Ma2u1 - Meu2), k == 4 && m == -1}, {-(Sqrt[21]*(2*Sqrt[5]*Ma2u1 - 15*Meu1 + Sqrt[5]*Meu2))/20, k == 4 && m == 0}, {(1/2 - I/2)*Sqrt[3/7]*(Ma2u1 - Meu2), k == 4 && m == 1}, {(-I)*Sqrt[6/7]*(Ma2u1 - Meu2), k == 4 && m == 2}}, (1/2 + I/2)*Sqrt[3]*(Ma2u1 - Meu2)] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D3d_111.Quanty> |
| | |
| | Akm = {{2, 1, (-1/3+1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , |
| | {2,-1, (1/3+1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , |
| | {2, 2, (-1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , |
| | {2,-2, (1/3*I)*((sqrt(5/14))*((3)*(Ma2u1) + (4)*(Meu2)))} , |
| | {4, 0, (-1/20)*((sqrt(21))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , |
| | {4,-1, (-1/2+-1/2*I)*((sqrt(3/7))*(Ma2u1 + (-1)*(Meu2)))} , |
| | {4, 1, (1/2+-1/2*I)*((sqrt(3/7))*(Ma2u1 + (-1)*(Meu2)))} , |
| | {4, 2, (-I)*((sqrt(6/7))*(Ma2u1 + (-1)*(Meu2)))} , |
| | {4,-2, (I)*((sqrt(6/7))*(Ma2u1 + (-1)*(Meu2)))} , |
| | {4,-3, (-1/2+1/2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , |
| | {4, 3, (1/2+1/2*I)*((sqrt(3))*(Ma2u1 + (-1)*(Meu2)))} , |
| | {4,-4, (-1/4)*((sqrt(3/10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} , |
| | {4, 4, (-1/4)*((sqrt(3/10))*((2)*((sqrt(5))*(Ma2u1)) + (-15)*(Meu1) + (sqrt(5))*(Meu2)))} } |
| | |
| | </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{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} $|$ (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right] $|$ \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} $|$ (2 \text{Ma2u1}+\text{Meu2}) \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right] $|$ -\frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} $|$ \frac{(-1)^{3/4} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} $|$ \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} $| |
| | ^$ {Y_{0}^{(1)}} $|$ \left(-\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) $|$ -\frac{i \text{Ma2u1}}{\sqrt{6}} $|$ (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right] $|$ -\frac{2 \text{Ma2u1}}{3 \sqrt{5}}+\text{Meu1}-\frac{\text{Meu2}}{3 \sqrt{5}} $|$ (\text{Ma2u1}+3 \text{Meu2}) \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right] $|$ \frac{i \text{Ma2u1}}{\sqrt{6}} $|$ \left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}-\text{Meu2}) $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{2 \text{Ma2u1}-3 \sqrt{5} \text{Meu1}+\text{Meu2}}{6 \sqrt{2}} $|$ \frac{\sqrt[4]{-1} (\text{Ma2u1}-\text{Meu2})}{\sqrt{6}} $|$ \frac{i (\text{Ma2u1}-2 \text{Meu2})}{\sqrt{30}} $|$ \frac{\left(\frac{1}{3}-\frac{i}{3}\right) (2 \text{Ma2u1}+\text{Meu2})}{\sqrt{10}} $|$ \frac{2 \sqrt{5} \text{Ma2u1}-15 \text{Meu1}+\sqrt{5} \text{Meu2}}{10 \sqrt{6}} $|$ (\text{Ma2u1}+\text{Meu2}) \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right] $|$ \frac{i (\text{Ma2u1}+2 \text{Meu2})}{3 \sqrt{2}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x^3+y^3+z^3} $ ^ $ f_{x^3-y^3} $ ^ $ f_{2z^3-x^3-y^3} $ ^ $ f_{\left(y^2-z^2\right)x+\left(z^2-x^2\right)y+\left(x^2-y^2\right)z} $ ^ $ f_{-\left(y^2-z^2\right)x-\left(z^2-x^2\right)y+2\left(x^2-y^2\right)z} $ ^ $ f_{-\left(y^2-z^2\right)x+\left(z^2-x^2\right)y} $ ^ |
| | ^$ p_{x+y+z} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(2-2 i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(-1+i)\right)\right) $|$ \left(-\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(-15 \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+15 i \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]-60 \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+(29+29 i) \sqrt{5}\right)+\text{Meu2} \left(-45 \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+45 i \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]-30 \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+(7+7 i) \sqrt{5}\right)-(90+90 i) \text{Meu1}\right) $|$ \frac{\left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)}{\sqrt{2}} $|$ \frac{\text{Ma2u1} \left(-(3-3 i) \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+(3+3 i) \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+(24-24 i) \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+2 \sqrt{5}\right)+(1+i) \text{Meu2} \left(9 i \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+9 \sqrt{3} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]-12 i \sqrt{2} \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+(-1+i) \sqrt{5}\right)}{36 \sqrt{2}} $|$ \left(-\frac{1}{12}-\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)\right) $|$ -\frac{\left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(4 \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+4 i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)+\text{Meu2} \left(4 \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+4 i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)\right)}{\sqrt{2}} $|$ \frac{\left(\frac{1}{12}+\frac{i}{12}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(i \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+(-1+i)\right)}{\sqrt{2}} $| |
| | ^$ p_{x-y} $|$ \frac{\left(\frac{1}{2}+\frac{i}{2}\right) (\text{Ma2u1}+\text{Meu2}) \left(\text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]\right)}{\sqrt{2}} $|$ \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+i \sqrt{10}\right)}{30 \sqrt{3}} $|$ \text{Meu1} $|$ \frac{(2 \text{Ma2u1}+\text{Meu2}) \left((15+15 i) \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+i \sqrt{10}\right)}{15 \sqrt{6}} $|$ \frac{\left(\frac{1}{6}+\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(-1+i)\right)}{\sqrt{2}} $|$ \left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(-1+i)\right)+\text{Meu2} \left(i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(2-2 i)\right)\right) $|$ 0 $| |
| | ^$ p_{3z-r} $|$ \frac{\left(\frac{1}{6}-\frac{i}{6}\right) (\text{Ma2u1}+\text{Meu2}) \left(\sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]-i \sqrt{3} \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+(1+i)\right)}{\sqrt{2}} $|$ \left(\frac{1}{180}-\frac{i}{180}\right) \left(\text{Ma2u1} \left(-4 \text{Root}\left[\text{$\#$1}^4+25\$|$,1\right]+15 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]-15 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+(1+i) \sqrt{10}\right)+\text{Meu2} \left(-2 \text{Root}\left[\text{$\#$1}^4+25\$|$,1\right]+45 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]-45 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+(8+8 i) \sqrt{10}\right)\right) $|$ \left(\frac{1}{4}+\frac{i}{4}\right) (\text{Ma2u1}+3 \text{Meu2}) \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right) $|$ \frac{\left(\frac{1}{180}+\frac{i}{180}\right) \left(\text{Ma2u1} \left(15 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+15 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+120 i \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+(-7+7 i) \sqrt{10}\right)+\text{Meu2} \left(45 i \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+45 \sqrt{6} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+60 i \text{Root}\left[2025 \text{$\#$1}^4+1\$|$,1\right]+(-11+11 i) \sqrt{10}\right)+(90-90 i) \sqrt{2} \text{Meu1}\right)}{\sqrt{2}} $|$ \frac{\left(\frac{1}{6}+\frac{i}{6}\right) \left(\text{Ma2u1} \left(\text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)+\text{Meu2} \left(\text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]+3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)\right)}{\sqrt{2}} $|$ \left(\frac{1}{12}+\frac{i}{12}\right) \left(\text{Ma2u1} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]-\sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)+\text{Meu2} \left(2 \text{Root}\left[36 \text{$\#$1}^4+1\$|$,1\right]+2 i \text{Root}\left[36 \text{$\#$1}^4+1\$|$,3\right]-3 \sqrt{5} \left(\text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+i \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]\right)\right)\right) $|$ \frac{1}{12} \left(\text{Ma2u1} \left(-(1-i) \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+(1+i) \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]-2\right)+(3+3 i) \text{Meu2} \left(i \sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,1\right]+\sqrt{15} \text{Root}\left[900 \text{$\#$1}^4+1\$|$,3\right]+(1-i)\right)\right) $| |
| | |
| | |
| | ### |
| | |
| | </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> | | | | | | |
| | |
| | ### |
