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| physics_chemistry:point_groups:d2h:orientation_xyz [2018/03/21 17:25] – created Stefano Agrestini | physics_chemistry:point_groups:d2h:orientation_xyz [2018/04/06 09:06] (current) – Maurits W. Haverkort |
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| | ~~CLOSETOC~~ |
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| ====== Orientation XYZ ====== | ====== Orientation XYZ ====== |
| | |
| | ===== Symmetry Operations ===== |
| |
| ### | ### |
| alligned paragraph text | |
| | In the D2h Point Group, with orientation XYZ there are the following symmetry operations |
| ### | ### |
| |
| ===== Example ===== | ### |
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| | {{:physics_chemistry:pointgroup:d2h_xyz.png}} |
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| ### | ### |
| description text | |
| ### | ### |
| |
| ==== Input ==== | ^ Operator ^ Orientation ^ |
| <code Quanty Example.Quanty> | ^ $\text{E}$ | $\{0,0,0\}$ , | |
| -- some example code | ^ $C_2$ | $\{0,0,1\}$ , | |
| | ^ $C_2$ | $\{0,1,0\}$ , | |
| | ^ $C_2$ | $\{1,0,0\}$ , | |
| | ^ $\text{i}$ | $\{0,0,0\}$ , | |
| | ^ $\sigma _h$ | $\{0,0,1\}$ , | |
| | ^ $\sigma _h$ | $\{0,1,0\}$ , | |
| | ^ $\sigma _h$ | $\{1,0,0\}$ , | |
| | |
| | ### |
| | |
| | ===== Different Settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]] |
| | |
| | ### |
| | |
| | ===== Character Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ $ \text{i} \,{\text{(1)}} $ ^ $ \sigma_h \,{\text{(1)}} $ ^ $ \sigma_h \,{\text{(1)}} $ ^ $ \sigma_h \,{\text{(1)}} $ ^ |
| | ^ $ A_g $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | |
| | ^ $ B_{1g} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | |
| | ^ $ B_{2g} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | |
| | ^ $ B_{3g} $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | |
| | ^ $ A_u $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | |
| | ^ $ B_{1u} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | |
| | ^ $ B_{2u} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | |
| | ^ $ B_{3u} $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | |
| | |
| | ### |
| | |
| | ===== Product Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ A_g $ ^ $ B_{1g} $ ^ $ B_{2g} $ ^ $ B_{3g} $ ^ $ A_u $ ^ $ B_{1u} $ ^ $ B_{2u} $ ^ $ B_{3u} $ ^ |
| | ^ $ A_g $ | $ A_g $ | $ B_{1g} $ | $ B_{2g} $ | $ B_{3g} $ | $ A_u $ | $ B_{1u} $ | $ B_{2u} $ | $ B_{3u} $ | |
| | ^ $ B_{1g} $ | $ B_{1g} $ | $ A_g $ | $ B_{3g} $ | $ B_{2g} $ | $ B_{1u} $ | $ A_u $ | $ B_{3u} $ | $ B_{2u} $ | |
| | ^ $ B_{2g} $ | $ B_{2g} $ | $ B_{3g} $ | $ A_g $ | $ B_{1g} $ | $ B_{2u} $ | $ B_{3u} $ | $ A_u $ | $ B_{1u} $ | |
| | ^ $ B_{3g} $ | $ B_{3g} $ | $ B_{2g} $ | $ B_{1g} $ | $ A_g $ | $ B_{3u} $ | $ B_{2u} $ | $ B_{1u} $ | $ A_u $ | |
| | ^ $ A_u $ | $ A_u $ | $ B_{1u} $ | $ B_{2u} $ | $ B_{3u} $ | $ A_g $ | $ B_{1g} $ | $ B_{2g} $ | $ B_{3g} $ | |
| | ^ $ B_{1u} $ | $ B_{1u} $ | $ A_u $ | $ B_{3u} $ | $ B_{2u} $ | $ B_{1g} $ | $ A_g $ | $ B_{3g} $ | $ B_{2g} $ | |
| | ^ $ B_{2u} $ | $ B_{2u} $ | $ B_{3u} $ | $ A_u $ | $ B_{1u} $ | $ B_{2g} $ | $ B_{3g} $ | $ A_g $ | $ B_{1g} $ | |
| | ^ $ B_{3u} $ | $ B_{3u} $ | $ B_{2u} $ | $ B_{1u} $ | $ A_u $ | $ B_{3g} $ | $ B_{2g} $ | $ B_{1g} $ | $ A_g $ | |
| | |
| | ### |
| | |
| | ===== Sub Groups with compatible settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] |
| | * [[physics_chemistry:point_groups:c2h:orientation_z|Point Group C2h with orientation Z]] |
| | * [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]] |
| | * [[physics_chemistry:point_groups:c2:orientation_y|Point Group C2 with orientation Y]] |
| | * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]] |
| | * [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]] |
| | * [[physics_chemistry:point_groups:cs:orientation_x|Point Group Cs with orientation X]] |
| | * [[physics_chemistry:point_groups:cs:orientation_y|Point Group Cs with orientation Y]] |
| | * [[physics_chemistry:point_groups:cs:orientation_z|Point Group Cs with orientation Z]] |
| | * [[physics_chemistry:point_groups:d2:orientation_xyz|Point Group D2 with orientation XYZ]] |
| | |
| | ### |
| | |
| | ===== Super Groups with compatible settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]] |
| | * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]] |
| | * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:th:orientation_xyz|Point Group Th with orientation xyz]] |
| | |
| | ### |
| | |
| | ===== Invariant Potential expanded on renormalized spherical Harmonics ===== |
| | |
| | ### |
| | |
| | Any potential (function) can be written as a sum over spherical harmonics. |
| | $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ |
| | Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ |
| | The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the D2h Point group with orientation XYZ the form of the expansion coefficients is: |
| | |
| | ### |
| | |
| | ==== Expansion ==== |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | A(2,2) & k=2\land (m=-2\lor m=2) \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(4,4) & k=4\land (m=-4\lor m=4) \\ |
| | A(4,2) & k=4\land (m=-2\lor m=2) \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(6,6) & k=6\land (m=-6\lor m=6) \\ |
| | A(6,4) & k=6\land (m=-4\lor m=4) \\ |
| | A(6,2) & k=6\land (m=-2\lor m=2) \\ |
| | A(6,0) & k=6\land m=0 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {A[4, 0], k == 4 && m == 0}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {A[6, 0], k == 6 && m == 0}}, 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_D2h_XYZ.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2)} , |
| | {2, 2, A(2,2)} , |
| | {4, 0, A(4,0)} , |
| | {4,-2, A(4,2)} , |
| | {4, 2, A(4,2)} , |
| | {4,-4, A(4,4)} , |
| | {4, 4, A(4,4)} , |
| | {6, 0, A(6,0)} , |
| | {6,-2, A(6,2)} , |
| | {6, 2, A(6,2)} , |
| | {6,-4, A(6,4)} , |
| | {6, 4, A(6,4)} , |
| | {6,-6, A(6,6)} , |
| | {6, 6, A(6,6)} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | ==== One particle coupling on a basis of spherical harmonics ==== |
| | |
| | ### |
| | |
| | The operator representing the potential in second quantisation is given as: |
| | $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| | For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. |
| | $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ |
| | Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$ |
| | |
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| | ### |
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| | ### |
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| | we can express the operator as |
| | $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| | |
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| | ### |
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| | ### |
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| | The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter. |
| | |
| | ### |
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| | ### |
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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{\text{Asd}(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{\text{Asd}(2,2)}{\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)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ 0 $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $| |
| | ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{\text{Asd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} \text{Add}(4,2)-\frac{2}{7} \text{Add}(2,2) $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\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 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$ 0 $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $| |
| | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ 0 $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $| |
| | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $| |
| | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ 0 $| |
| | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,2)-\frac{1}{3} \sqrt{\frac{5}{7}} \text{Apf}(4,2) $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)-\frac{2}{33} \sqrt{10} \text{Aff}(4,2)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $| |
| | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2,2)+\frac{2 \text{Apf}(4,2)}{3 \sqrt{7}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ -\frac{2 \text{Aff}(2,2)}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} \text{Aff}(6,2) $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| |
| | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ 0 $|$ \frac{3 \text{Apf}(2,2)}{\sqrt{35}}-\frac{\text{Apf}(4,2)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{11} \sqrt{6} \text{Aff}(4,2)-\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| |
| | |
| | |
| | ### |
| | |
| | ==== 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 $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | ==== One particle coupling on a basis of symmetry adapted functions ==== |
| | |
| | ### |
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| | After rotation we find |
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| | ### |
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| | |
| | ### |
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| | | $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ p_x $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)+\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$ 0 $| |
| | ^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $| |
| | ^$ d_{x^2-y^2} $|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ 0 $|$ 0 $|$ 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} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 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{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 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 }$|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ 0 $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ 0 $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $| |
| | |
| | |
| | ### |
| | |
| | ===== 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_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.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:d2h_xyz_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** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{3} (\text{Eb1u}+\text{Eb2u}+\text{Eb3u}) & k=0\land m=0 \\ |
| | -\frac{5 (\text{Eb2u}-\text{Eb3u})}{2 \sqrt{6}} & k=2\land (m=-2\lor m=2) \\ |
| | \frac{5}{6} (2 \text{Eb1u}-\text{Eb2u}-\text{Eb3u}) & k=2\land m=0 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eb1u + Eb2u + Eb3u)/3, k == 0 && m == 0}, {(-5*(Eb2u - Eb3u))/(2*Sqrt[6]), k == 2 && (m == -2 || m == 2)}, {(5*(2*Eb1u - Eb2u - Eb3u))/6, k == 2 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty> |
| | |
| | Akm = {{0, 0, (1/3)*(Eb1u + Eb2u + Eb3u)} , |
| | {2, 0, (5/6)*((2)*(Eb1u) + (-1)*(Eb2u) + (-1)*(Eb3u))} , |
| | {2,-2, (-5/2)*((1/(sqrt(6)))*(Eb2u + (-1)*(Eb3u)))} , |
| | {2, 2, (-5/2)*((1/(sqrt(6)))*(Eb2u + (-1)*(Eb3u)))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{\text{Eb2u}+\text{Eb3u}}{2} $|$ 0 $|$ \frac{\text{Eb2u}-\text{Eb3u}}{2} $| |
| | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Eb1u} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{\text{Eb2u}-\text{Eb3u}}{2} $|$ 0 $|$ \frac{\text{Eb2u}+\text{Eb3u}}{2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ |
| | ^$ p_x $|$ \text{Eb3u} $|$ 0 $|$ 0 $| |
| | ^$ p_y $|$ 0 $|$ \text{Eb2u} $|$ 0 $| |
| | ^$ p_z $|$ 0 $|$ 0 $|$ \text{Eb1u} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| |
| | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| |
| | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eb3u}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_1_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | |
| | ^ ^$$\text{Eb2u}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_1_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | |
| | ^ ^$$\text{Eb1u}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_1_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{5} (\text{Eagx2y2}+\text{Eagz2}+\text{Eb1g}+\text{Eb2g}+\text{Eb3g}) & k=0\land m=0 \\ |
| | \frac{1}{4} \left(\sqrt{6} \text{Eb2g}-\sqrt{6} \text{Eb3g}-4 \sqrt{2} \text{Mag}\right) & k=2\land (m=-2\lor m=2) \\ |
| | \frac{1}{2} (-2 \text{Eagx2y2}+2 \text{Eagz2}-2 \text{Eb1g}+\text{Eb2g}+\text{Eb3g}) & k=2\land m=0 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eagx2y2}-\text{Eb1g}) & k=4\land (m=-4\lor m=4) \\ |
| | \frac{3 \left(\text{Eb2g}-\text{Eb3g}+\sqrt{3} \text{Mag}\right)}{\sqrt{10}} & k=4\land (m=-2\lor m=2) \\ |
| | \frac{3}{10} (\text{Eagx2y2}+6 \text{Eagz2}+\text{Eb1g}-4 \text{Eb2g}-4 \text{Eb3g}) & k=4\land m=0 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eagx2y2 + Eagz2 + Eb1g + Eb2g + Eb3g)/5, k == 0 && m == 0}, {(Sqrt[6]*Eb2g - Sqrt[6]*Eb3g - 4*Sqrt[2]*Mag)/4, k == 2 && (m == -2 || m == 2)}, {(-2*Eagx2y2 + 2*Eagz2 - 2*Eb1g + Eb2g + Eb3g)/2, k == 2 && m == 0}, {(3*Sqrt[7/10]*(Eagx2y2 - Eb1g))/2, k == 4 && (m == -4 || m == 4)}, {(3*(Eb2g - Eb3g + Sqrt[3]*Mag))/Sqrt[10], k == 4 && (m == -2 || m == 2)}, {(3*(Eagx2y2 + 6*Eagz2 + Eb1g - 4*Eb2g - 4*Eb3g))/10, k == 4 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty> |
| | |
| | Akm = {{0, 0, (1/5)*(Eagx2y2 + Eagz2 + Eb1g + Eb2g + Eb3g)} , |
| | {2, 0, (1/2)*((-2)*(Eagx2y2) + (2)*(Eagz2) + (-2)*(Eb1g) + Eb2g + Eb3g)} , |
| | {2,-2, (1/4)*((sqrt(6))*(Eb2g) + (-1)*((sqrt(6))*(Eb3g)) + (-4)*((sqrt(2))*(Mag)))} , |
| | {2, 2, (1/4)*((sqrt(6))*(Eb2g) + (-1)*((sqrt(6))*(Eb3g)) + (-4)*((sqrt(2))*(Mag)))} , |
| | {4, 0, (3/10)*(Eagx2y2 + (6)*(Eagz2) + Eb1g + (-4)*(Eb2g) + (-4)*(Eb3g))} , |
| | {4,-2, (3)*((1/(sqrt(10)))*(Eb2g + (-1)*(Eb3g) + (sqrt(3))*(Mag)))} , |
| | {4, 2, (3)*((1/(sqrt(10)))*(Eb2g + (-1)*(Eb3g) + (sqrt(3))*(Mag)))} , |
| | {4,-4, (3/2)*((sqrt(7/10))*(Eagx2y2 + (-1)*(Eb1g)))} , |
| | {4, 4, (3/2)*((sqrt(7/10))*(Eagx2y2 + (-1)*(Eb1g)))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Eagx2y2}+\text{Eb1g}}{2} $|$ 0 $|$ \frac{\text{Mag}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Eagx2y2}-\text{Eb1g}}{2} $| |
| | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{\text{Eb2g}+\text{Eb3g}}{2} $|$ 0 $|$ \frac{\text{Eb3g}-\text{Eb2g}}{2} $|$ 0 $| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Mag}}{\sqrt{2}} $|$ 0 $|$ \text{Eagz2} $|$ 0 $|$ \frac{\text{Mag}}{\sqrt{2}} $| |
| | ^$ {Y_{1}^{(2)}} $|$ 0 $|$ \frac{\text{Eb3g}-\text{Eb2g}}{2} $|$ 0 $|$ \frac{\text{Eb2g}+\text{Eb3g}}{2} $|$ 0 $| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{\text{Eagx2y2}-\text{Eb1g}}{2} $|$ 0 $|$ \frac{\text{Mag}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Eagx2y2}+\text{Eb1g}}{2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ d_{x^2-y^2} $|$ \text{Eagx2y2} $|$ \text{Mag} $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ d_{3z^2-r^2} $|$ \text{Mag} $|$ \text{Eagz2} $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Eb3g} $|$ 0 $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb2g} $|$ 0 $| |
| | ^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb1g} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| |
| | ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eagx2y2}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_2_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | |
| | ^ ^$$\text{Eagz2}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_2_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eb3g}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_2_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | |
| | ^ ^$$\text{Eb2g}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_2_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | |
| | ^ ^$$\text{Eb1g}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_2_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for f orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{7} (\text{Eau}+\text{Eb1u1}+\text{Eb1u2}+\text{Eb2u1}+\text{Eb2u2}+\text{Eb3u1}+\text{Eb3u2}) & k=0\land m=0 \\ |
| | \frac{5}{28} \left(-\sqrt{6} \text{Eb2u1}+\sqrt{6} \text{Eb3u1}+\sqrt{10} (-2 \text{Mb1u}+\text{Mb2u}+\text{Mb3u})\right) & k=2\land (m=-2\lor m=2) \\ |
| | \frac{5}{14} \left(2 \text{Eb1u1}-\text{Eb2u1}-\text{Eb3u1}-\sqrt{15} \text{Mb2u}+\sqrt{15} \text{Mb3u}\right) & k=2\land m=0 \\ |
| | \frac{3 \left(-4 \sqrt{5} \text{Eau}+4 \sqrt{5} \text{Eb1u2}+3 \sqrt{5} \text{Eb2u1}-3 \sqrt{5} \text{Eb2u2}+3 \sqrt{5} \text{Eb3u1}-3 \sqrt{5} \text{Eb3u2}-2 \sqrt{3} \text{Mb2u}+2 \sqrt{3} \text{Mb3u}\right)}{8 \sqrt{14}} & k=4\land (m=-4\lor m=4) \\ |
| | \frac{3}{56} \left(3 \sqrt{10} \text{Eb2u1}-7 \sqrt{10} \text{Eb2u2}-3 \sqrt{10} \text{Eb3u1}+7 \sqrt{10} \text{Eb3u2}-4 \sqrt{6} \text{Mb1u}+2 \sqrt{6} \text{Mb2u}+2 \sqrt{6} \text{Mb3u}\right) & k=4\land (m=-2\lor m=2) \\ |
| | \frac{3}{56} \left(-28 \text{Eau}+24 \text{Eb1u1}-28 \text{Eb1u2}+9 \text{Eb2u1}+7 \text{Eb2u2}+9 \text{Eb3u1}+7 \text{Eb3u2}+2 \sqrt{15} \text{Mb2u}-2 \sqrt{15} \text{Mb3u}\right) & k=4\land m=0 \\ |
| | -\frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eb2u1}+3 \sqrt{3} \text{Eb2u2}-5 \sqrt{3} \text{Eb3u1}-3 \sqrt{3} \text{Eb3u2}+6 \sqrt{5} \text{Mb2u}+6 \sqrt{5} \text{Mb3u}\right) & k=6\land (m=-6\lor m=6) \\ |
| | -\frac{13 \left(24 \text{Eau}-24 \text{Eb1u2}+15 \text{Eb2u1}-15 \text{Eb2u2}+15 \text{Eb3u1}-15 \text{Eb3u2}-2 \sqrt{15} \text{Mb2u}+2 \sqrt{15} \text{Mb3u}\right)}{80 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ |
| | -\frac{13 \left(5 \sqrt{15} \text{Eb2u1}+3 \sqrt{15} \text{Eb2u2}-5 \sqrt{15} \text{Eb3u1}-3 \sqrt{15} \text{Eb3u2}-64 \text{Mb1u}-34 \text{Mb2u}-34 \text{Mb3u}\right)}{160 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\ |
| | \frac{13}{560} \left(24 \text{Eau}+80 \text{Eb1u1}+24 \text{Eb1u2}-25 \text{Eb2u1}-39 \text{Eb2u2}-25 \text{Eb3u1}-39 \text{Eb3u2}+14 \sqrt{15} \text{Mb2u}-14 \sqrt{15} \text{Mb3u}\right) & k=6\land m=0 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eau + Eb1u1 + Eb1u2 + Eb2u1 + Eb2u2 + Eb3u1 + Eb3u2)/7, k == 0 && m == 0}, {(5*(-(Sqrt[6]*Eb2u1) + Sqrt[6]*Eb3u1 + Sqrt[10]*(-2*Mb1u + Mb2u + Mb3u)))/28, k == 2 && (m == -2 || m == 2)}, {(5*(2*Eb1u1 - Eb2u1 - Eb3u1 - Sqrt[15]*Mb2u + Sqrt[15]*Mb3u))/14, k == 2 && m == 0}, {(3*(-4*Sqrt[5]*Eau + 4*Sqrt[5]*Eb1u2 + 3*Sqrt[5]*Eb2u1 - 3*Sqrt[5]*Eb2u2 + 3*Sqrt[5]*Eb3u1 - 3*Sqrt[5]*Eb3u2 - 2*Sqrt[3]*Mb2u + 2*Sqrt[3]*Mb3u))/(8*Sqrt[14]), k == 4 && (m == -4 || m == 4)}, {(3*(3*Sqrt[10]*Eb2u1 - 7*Sqrt[10]*Eb2u2 - 3*Sqrt[10]*Eb3u1 + 7*Sqrt[10]*Eb3u2 - 4*Sqrt[6]*Mb1u + 2*Sqrt[6]*Mb2u + 2*Sqrt[6]*Mb3u))/56, k == 4 && (m == -2 || m == 2)}, {(3*(-28*Eau + 24*Eb1u1 - 28*Eb1u2 + 9*Eb2u1 + 7*Eb2u2 + 9*Eb3u1 + 7*Eb3u2 + 2*Sqrt[15]*Mb2u - 2*Sqrt[15]*Mb3u))/56, k == 4 && m == 0}, {(-13*Sqrt[11/7]*(5*Sqrt[3]*Eb2u1 + 3*Sqrt[3]*Eb2u2 - 5*Sqrt[3]*Eb3u1 - 3*Sqrt[3]*Eb3u2 + 6*Sqrt[5]*Mb2u + 6*Sqrt[5]*Mb3u))/160, k == 6 && (m == -6 || m == 6)}, {(-13*(24*Eau - 24*Eb1u2 + 15*Eb2u1 - 15*Eb2u2 + 15*Eb3u1 - 15*Eb3u2 - 2*Sqrt[15]*Mb2u + 2*Sqrt[15]*Mb3u))/(80*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(-13*(5*Sqrt[15]*Eb2u1 + 3*Sqrt[15]*Eb2u2 - 5*Sqrt[15]*Eb3u1 - 3*Sqrt[15]*Eb3u2 - 64*Mb1u - 34*Mb2u - 34*Mb3u))/(160*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {(13*(24*Eau + 80*Eb1u1 + 24*Eb1u2 - 25*Eb2u1 - 39*Eb2u2 - 25*Eb3u1 - 39*Eb3u2 + 14*Sqrt[15]*Mb2u - 14*Sqrt[15]*Mb3u))/560, k == 6 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty> |
| | |
| | Akm = {{0, 0, (1/7)*(Eau + Eb1u1 + Eb1u2 + Eb2u1 + Eb2u2 + Eb3u1 + Eb3u2)} , |
| | {2, 0, (5/14)*((2)*(Eb1u1) + (-1)*(Eb2u1) + (-1)*(Eb3u1) + (-1)*((sqrt(15))*(Mb2u)) + (sqrt(15))*(Mb3u))} , |
| | {2,-2, (5/28)*((-1)*((sqrt(6))*(Eb2u1)) + (sqrt(6))*(Eb3u1) + (sqrt(10))*((-2)*(Mb1u) + Mb2u + Mb3u))} , |
| | {2, 2, (5/28)*((-1)*((sqrt(6))*(Eb2u1)) + (sqrt(6))*(Eb3u1) + (sqrt(10))*((-2)*(Mb1u) + Mb2u + Mb3u))} , |
| | {4, 0, (3/56)*((-28)*(Eau) + (24)*(Eb1u1) + (-28)*(Eb1u2) + (9)*(Eb2u1) + (7)*(Eb2u2) + (9)*(Eb3u1) + (7)*(Eb3u2) + (2)*((sqrt(15))*(Mb2u)) + (-2)*((sqrt(15))*(Mb3u)))} , |
| | {4,-2, (3/56)*((3)*((sqrt(10))*(Eb2u1)) + (-7)*((sqrt(10))*(Eb2u2)) + (-3)*((sqrt(10))*(Eb3u1)) + (7)*((sqrt(10))*(Eb3u2)) + (-4)*((sqrt(6))*(Mb1u)) + (2)*((sqrt(6))*(Mb2u)) + (2)*((sqrt(6))*(Mb3u)))} , |
| | {4, 2, (3/56)*((3)*((sqrt(10))*(Eb2u1)) + (-7)*((sqrt(10))*(Eb2u2)) + (-3)*((sqrt(10))*(Eb3u1)) + (7)*((sqrt(10))*(Eb3u2)) + (-4)*((sqrt(6))*(Mb1u)) + (2)*((sqrt(6))*(Mb2u)) + (2)*((sqrt(6))*(Mb3u)))} , |
| | {4,-4, (3/8)*((1/(sqrt(14)))*((-4)*((sqrt(5))*(Eau)) + (4)*((sqrt(5))*(Eb1u2)) + (3)*((sqrt(5))*(Eb2u1)) + (-3)*((sqrt(5))*(Eb2u2)) + (3)*((sqrt(5))*(Eb3u1)) + (-3)*((sqrt(5))*(Eb3u2)) + (-2)*((sqrt(3))*(Mb2u)) + (2)*((sqrt(3))*(Mb3u))))} , |
| | {4, 4, (3/8)*((1/(sqrt(14)))*((-4)*((sqrt(5))*(Eau)) + (4)*((sqrt(5))*(Eb1u2)) + (3)*((sqrt(5))*(Eb2u1)) + (-3)*((sqrt(5))*(Eb2u2)) + (3)*((sqrt(5))*(Eb3u1)) + (-3)*((sqrt(5))*(Eb3u2)) + (-2)*((sqrt(3))*(Mb2u)) + (2)*((sqrt(3))*(Mb3u))))} , |
| | {6, 0, (13/560)*((24)*(Eau) + (80)*(Eb1u1) + (24)*(Eb1u2) + (-25)*(Eb2u1) + (-39)*(Eb2u2) + (-25)*(Eb3u1) + (-39)*(Eb3u2) + (14)*((sqrt(15))*(Mb2u)) + (-14)*((sqrt(15))*(Mb3u)))} , |
| | {6,-2, (-13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eb2u1)) + (3)*((sqrt(15))*(Eb2u2)) + (-5)*((sqrt(15))*(Eb3u1)) + (-3)*((sqrt(15))*(Eb3u2)) + (-64)*(Mb1u) + (-34)*(Mb2u) + (-34)*(Mb3u)))} , |
| | {6, 2, (-13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eb2u1)) + (3)*((sqrt(15))*(Eb2u2)) + (-5)*((sqrt(15))*(Eb3u1)) + (-3)*((sqrt(15))*(Eb3u2)) + (-64)*(Mb1u) + (-34)*(Mb2u) + (-34)*(Mb3u)))} , |
| | {6,-4, (-13/80)*((1/(sqrt(14)))*((24)*(Eau) + (-24)*(Eb1u2) + (15)*(Eb2u1) + (-15)*(Eb2u2) + (15)*(Eb3u1) + (-15)*(Eb3u2) + (-2)*((sqrt(15))*(Mb2u)) + (2)*((sqrt(15))*(Mb3u))))} , |
| | {6, 4, (-13/80)*((1/(sqrt(14)))*((24)*(Eau) + (-24)*(Eb1u2) + (15)*(Eb2u1) + (-15)*(Eb2u2) + (15)*(Eb3u1) + (-15)*(Eb3u2) + (-2)*((sqrt(15))*(Mb2u)) + (2)*((sqrt(15))*(Mb3u))))} , |
| | {6,-6, (-13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eb2u1)) + (3)*((sqrt(3))*(Eb2u2)) + (-5)*((sqrt(3))*(Eb3u1)) + (-3)*((sqrt(3))*(Eb3u2)) + (6)*((sqrt(5))*(Mb2u)) + (6)*((sqrt(5))*(Mb3u))))} , |
| | {6, 6, (-13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eb2u1)) + (3)*((sqrt(3))*(Eb2u2)) + (-5)*((sqrt(3))*(Eb3u1)) + (-3)*((sqrt(3))*(Eb3u2)) + (6)*((sqrt(5))*(Mb2u)) + (6)*((sqrt(5))*(Mb3u))))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{-3}^{(3)}} $|$ \frac{1}{16} \left(5 \text{Eb2u1}+3 \text{Eb2u2}+5 \text{Eb3u1}+3 \text{Eb3u2}+2 \sqrt{15} (\text{Mb2u}-\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}-\sqrt{15} \text{Eb3u1}+\sqrt{15} \text{Eb3u2}-2 (\text{Mb2u}+\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}+\sqrt{15} \text{Eb3u1}-\sqrt{15} \text{Eb3u2}-2 \text{Mb2u}+2 \text{Mb3u}\right) $|$ 0 $|$ \frac{1}{16} \left(5 \text{Eb2u1}+3 \text{Eb2u2}-5 \text{Eb3u1}-3 \text{Eb3u2}+2 \sqrt{15} (\text{Mb2u}+\text{Mb3u})\right) $| |
| | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Eau}+\text{Eb1u2}}{2} $|$ 0 $|$ \frac{\text{Mb1u}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Eb1u2}-\text{Eau}}{2} $|$ 0 $| |
| | ^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}-\sqrt{15} \text{Eb3u1}+\sqrt{15} \text{Eb3u2}-2 (\text{Mb2u}+\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eb2u1}+5 \text{Eb2u2}+3 \text{Eb3u1}+5 \text{Eb3u2}+2 \sqrt{15} (\text{Mb3u}-\text{Mb2u})\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eb2u1}+5 \text{Eb2u2}-3 \text{Eb3u1}-5 \text{Eb3u2}-2 \sqrt{15} (\text{Mb2u}+\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}+\sqrt{15} \text{Eb3u1}-\sqrt{15} \text{Eb3u2}-2 \text{Mb2u}+2 \text{Mb3u}\right) $| |
| | ^$ {Y_{0}^{(3)}} $|$ 0 $|$ \frac{\text{Mb1u}}{\sqrt{2}} $|$ 0 $|$ \text{Eb1u1} $|$ 0 $|$ \frac{\text{Mb1u}}{\sqrt{2}} $|$ 0 $| |
| | ^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}+\sqrt{15} \text{Eb3u1}-\sqrt{15} \text{Eb3u2}-2 \text{Mb2u}+2 \text{Mb3u}\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eb2u1}+5 \text{Eb2u2}-3 \text{Eb3u1}-5 \text{Eb3u2}-2 \sqrt{15} (\text{Mb2u}+\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eb2u1}+5 \text{Eb2u2}+3 \text{Eb3u1}+5 \text{Eb3u2}+2 \sqrt{15} (\text{Mb3u}-\text{Mb2u})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}-\sqrt{15} \text{Eb3u1}+\sqrt{15} \text{Eb3u2}-2 (\text{Mb2u}+\text{Mb3u})\right) $| |
| | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{\text{Eb1u2}-\text{Eau}}{2} $|$ 0 $|$ \frac{\text{Mb1u}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Eau}+\text{Eb1u2}}{2} $|$ 0 $| |
| | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{16} \left(5 \text{Eb2u1}+3 \text{Eb2u2}-5 \text{Eb3u1}-3 \text{Eb3u2}+2 \sqrt{15} (\text{Mb2u}+\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}+\sqrt{15} \text{Eb3u1}-\sqrt{15} \text{Eb3u2}-2 \text{Mb2u}+2 \text{Mb3u}\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eb2u1}-\sqrt{15} \text{Eb2u2}-\sqrt{15} \text{Eb3u1}+\sqrt{15} \text{Eb3u2}-2 (\text{Mb2u}+\text{Mb3u})\right) $|$ 0 $|$ \frac{1}{16} \left(5 \text{Eb2u1}+3 \text{Eb2u2}+5 \text{Eb3u1}+3 \text{Eb3u2}+2 \sqrt{15} (\text{Mb2u}-\text{Mb3u})\right) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ f_{\text{xyz}} $|$ \text{Eau} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Eb3u1} $|$ 0 $|$ 0 $|$ \text{Mb3u} $|$ 0 $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Eb2u1} $|$ 0 $|$ 0 $|$ \text{Mb2u} $|$ 0 $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb1u1} $|$ 0 $|$ 0 $|$ \text{Mb1u} $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \text{Mb3u} $|$ 0 $|$ 0 $|$ \text{Eb3u2} $|$ 0 $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ 0 $|$ \text{Mb2u} $|$ 0 $|$ 0 $|$ \text{Eb2u2} $|$ 0 $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb1u} $|$ 0 $|$ 0 $|$ \text{Eb1u2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eau}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | |
| | ^ ^$$\text{Eb3u1}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\text{Eb2u1}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\text{Eb1u1}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | |
| | ^ ^$$\text{Eb3u2}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eb2u2}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_6.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eb1u2}$$ | {{:physics_chemistry:pointgroup:d2h_xyz_orb_3_7.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ===== Coupling between two shells ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| | |
| | ==== Potential for s-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| | A(2,2) & k=2\land (m=-2\lor m=2) \\ |
| | A(2,0) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != 0 && m != 2)}, {A[2, 2], k == 2 && (m == -2 || m == 2)}}, A[2, 0]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty> |
| | |
| | Akm = {{2, 0, A(2,0)} , |
| | {2,-2, A(2,2)} , |
| | {2, 2, A(2,2)} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \frac{A(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,2)}{\sqrt{5}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ \text{s} $|$ \sqrt{\frac{2}{5}} A(2,2) $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k=0\land m=0 \\ |
| | A(2,2) & k=2\land (m=-2\lor m=2) \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(4,4) & k=4\land (m=-4\lor m=4) \\ |
| | A(4,2) & k=4\land (m=-2\lor m=2) \\ |
| | A(4,0) & k=4\land m=0 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 2], k == 2 && (m == -2 || m == 2)}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 2], k == 4 && (m == -2 || m == 2)}, {A[4, 0], k == 4 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_D2h_XYZ.Quanty> |
| | |
| | Akm = {{2, 0, A(2,0)} , |
| | {2,-2, A(2,2)} , |
| | {2, 2, A(2,2)} , |
| | {4, 0, A(4,0)} , |
| | {4,-2, A(4,2)} , |
| | {4, 2, A(4,2)} , |
| | {4,-4, A(4,4)} , |
| | {4, 4, A(4,4)} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ 0 $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} A(2,2)+\frac{2 A(4,2)}{3 \sqrt{7}} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{105} \left(3 \sqrt{21} A(2,2)-5 \sqrt{35} A(4,2)\right) $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{3 A(2,2)}{\sqrt{35}}-\frac{A(4,2)}{3 \sqrt{21}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ p_x $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ 0 $|$ 0 $|$ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ 0 $|$ 0 $| |
| | ^$ p_y $|$ 0 $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ 0 $|$ 0 $|$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ 0 $| |
| | ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | |
| | ===== 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> | | | | | | |
| | |
| | ### |
