| Both sides previous revisionPrevious revision | |
| physics_chemistry:point_groups:c2:orientation_z [2018/03/22 23:24] – Stefano Agrestini | physics_chemistry:point_groups:c2:orientation_z [2018/04/06 09:13] (current) – Maurits W. Haverkort |
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| | ~~CLOSETOC~~ |
| | |
| ====== Orientation Z ====== | ====== Orientation Z ====== |
| | |
| | ===== Symmetry Operations ===== |
| |
| ### | ### |
| ====== Character table ====== | |
| ^C<sub>2</sub> | $E$ | $C$<sub>2</sub> | | In the C2 Point Group, with orientation Z there are the following symmetry operations |
| ^A | +1 | +1 | | |
| ^B | +1 | -1 | | |
| |
| ### | ### |
| |
| ===== Example ===== | ### |
| | |
| | {{:physics_chemistry:pointgroup:c2_z.png}} |
| |
| ### | ### |
| description text | |
| ### | ### |
| |
| ==== Input ==== | ^ Operator ^ Orientation ^ |
| <code Quanty Example.Quanty> | ^ $\text{E}$ | $\{0,0,0\}$ , | |
| -- some example code | ^ $C_2$ | $\{0,0,1\}$ , | |
| | |
| | ### |
| | |
| | ===== Different Settings ===== |
| | |
| | ### |
| | |
| | * [[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]] |
| | |
| | ### |
| | |
| | ===== Character Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ |
| | ^ $ \text{A} $ | $ 1 $ | $ 1 $ | |
| | ^ $ \text{B} $ | $ 1 $ | $ -1 $ | |
| | |
| | ### |
| | |
| | ===== Product Table ===== |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{A} $ ^ $ \text{B} $ ^ |
| | ^ $ \text{A} $ | $ \text{A} $ | $ \text{B} $ | |
| | ^ $ \text{B} $ | $ \text{B} $ | $ \text{A} $ | |
| | |
| | ### |
| | |
| | ===== Sub Groups with compatible settings ===== |
| | |
| | ### |
| | |
| | * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] |
| | |
| | ### |
| | |
| | ===== Super Groups with compatible settings ===== |
| | |
| | ### |
| | |
| | * [[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:c4h:orientation_z|Point Group C4h with orientation Z]] |
| | * [[physics_chemistry:point_groups:c4v:orientation_zxy|Point Group C4v with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:c4:orientation_z|Point Group C4 with orientation Z]] |
| | * [[physics_chemistry:point_groups:c6h:orientation_z|Point Group C6h with orientation Z]] |
| | * [[physics_chemistry:point_groups:c6v:orientation_zx|Point Group C6v with orientation Zx]] |
| | * [[physics_chemistry:point_groups:c6v:orientation_zy|Point Group C6v with orientation Zy]] |
| | * [[physics_chemistry:point_groups:c6:orientation_z|Point Group C6 with orientation Z]] |
| | * [[physics_chemistry:point_groups:d2d:orientation_zxy|Point Group D2d with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:d2:orientation_xyz|Point Group D2 with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:d4d:orientation_zxy|Point Group D4d with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:d4:orientation_zxy|Point Group D4 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:d6:orientation_zxy|Point Group D6 with orientation Zxy]] |
| | * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:o:orientation_xyz|Point Group O with orientation XYZ]] |
| | * [[physics_chemistry:point_groups:s4:orientation_z|Point Group S4 with orientation Z]] |
| | * [[physics_chemistry:point_groups:td:orientation_xyz|Point Group Td with orientation xyz]] |
| | * [[physics_chemistry:point_groups:th:orientation_xyz|Point Group Th with orientation xyz]] |
| | * [[physics_chemistry:point_groups:t:orientation_xyz|Point Group T 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 C2 Point group with orientation Z the form of the expansion coefficients is: |
| | |
| | ### |
| | |
| | ==== Expansion ==== |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | A(3,2)-i B(3,2) & k=3\land m=-2 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,2)+i B(3,2) & k=3\land m=2 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | A(5,4)-i B(5,4) & k=5\land m=-4 \\ |
| | A(5,2)-i B(5,2) & k=5\land m=-2 \\ |
| | A(5,0) & k=5\land m=0 \\ |
| | A(5,2)+i B(5,2) & k=5\land m=2 \\ |
| | A(5,4)+i B(5,4) & k=5\land m=4 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {A[5, 0], k == 5 && m == 0}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}, {A[5, 4] + I*B[5, 4], k == 5 && m == 4}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| </code> | </code> |
| |
| ==== Result ==== | ### |
| <WRAP center box 100%> | |
| text produced as output | |
| </WRAP> | |
| |
| ===== Table of contents ===== | ==== Input format suitable for Quanty ==== |
| {{indexmenu>.#1}} | |
| |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1, 0, A(1,0)} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3, 0, A(3,0)} , |
| | {3,-2, A(3,2) + (-I)*(B(3,2))} , |
| | {3, 2, A(3,2) + (I)*(B(3,2))} , |
| | {4, 0, A(4,0)} , |
| | {4,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5, 0, A(5,0)} , |
| | {5,-2, A(5,2) + (-I)*(B(5,2))} , |
| | {5, 2, A(5,2) + (I)*(B(5,2))} , |
| | {5,-4, A(5,4) + (-I)*(B(5,4))} , |
| | {5, 4, A(5,4) + (I)*(B(5,4))} , |
| | {6, 0, A(6,0)} , |
| | {6,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(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 $A_{n''l'',n'l'}(k,m)$ |
| | |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | |
| | we can express the operator as |
| | $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| | |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | |
| | 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. |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | | $ $ ^ $ {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}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ 0 }$|$ \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,2)+i \text{Bsf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,2)-i \text{Bsf}(3,2)}{\sqrt{7}} }$|$\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)-i \text{Bpp}(2,2)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$ \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $|$ 0 $|$ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$ 0 $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} (\text{App}(2,2)+i \text{Bpp}(2,2)) $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $| |
| | ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$\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)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)-i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)-i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)-i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)-i \text{Bdd}(4,2)) $|$ 0 $|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)+i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)-i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)-i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ 0 }$|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$ 0 $|$ -\frac{1}{7} \sqrt{6} (\text{Add}(2,2)+i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)+i \text{Bdd}(4,2)) $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)+i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)+i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)-i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4)) $|$ 0 $|$ \frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2)) $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)+i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)+i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$| |
| | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}} $|$ 0 $|$ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)-i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$|$\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)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $| |
| | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)-i \text{Bsf}(3,2)}{\sqrt{7}} }$|$ 0 $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)-i \text{Bdf}(5,4)) }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ 0 $|$ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $|$ 0 $| |
| | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)-i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)-i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)-i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $| |
| | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)+i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)+i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)-i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)-i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$ 0 $|$ -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ -\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $|$ 0 $| |
| | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2)) $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)+i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)+i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $|$ 0 $|$ -\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)+i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2)) $| |
| | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)+i \text{Bsf}(3,2)}{\sqrt{7}} }$|$ 0 $|$ \sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}} $|$ 0 $|$\color{darkred}{ \frac{1}{11} \sqrt{10} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$ 0 $|$ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $|$ 0 $|$ -\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| |
| | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$ 0 $|$ \frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{2}{11} \sqrt{\frac{5}{3}} (\text{Adf}(5,4)+i \text{Bdf}(5,4)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)+i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4)) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2)) $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| |
| | |
| | |
| | ### |
| | |
| | ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== |
| | |
| | ### |
| | |
| | |
| | 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 ==== |
| | |
| | ### |
| | |
| | After rotation we find |
| | |
| | ### |
| | |
| | |
| | |
| | ### |
| | |
| | | $ $ ^ $ \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}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$| |
| | ^$ 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) $|$ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ 0 $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_y $|$\color{darkred}{ 0 }$|$ -\frac{1}{5} \sqrt{6} \text{Bpp}(2,2) $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0)-\frac{1}{5} \sqrt{6} \text{App}(2,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_z $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ 0 $|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $| |
| | ^$ d_{x^2-y^2} $|$ \sqrt{\frac{2}{5}} \text{Asd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$| |
| | ^$ d_{3z^2-r^2} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$ \frac{1}{7} \sqrt{\frac{10}{3}} \text{Add}(4,2)-\frac{2}{7} \sqrt{2} \text{Add}(2,2) $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$| |
| | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}-\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)-\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}-\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}}+\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{7} \sqrt{6} \text{Bdd}(2,2)-\frac{2}{21} \sqrt{10} \text{Bdd}(4,2) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)+\frac{1}{7} \sqrt{6} \text{Add}(2,2)-\frac{4}{21} \text{Add}(4,0)+\frac{2}{21} \sqrt{10} \text{Add}(4,2) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}+\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$| |
| | ^$ d_{\text{xy}} $|$ -\sqrt{\frac{2}{5}} \text{Bsd}(2,2) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$ \frac{2}{7} \sqrt{2} \text{Bdd}(2,2)-\frac{1}{7} \sqrt{\frac{10}{3}} \text{Bdd}(4,2) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$| |
| | ^$ f_{\text{xyz}} $|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$|$ 0 $|$ 0 $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Bpf}(4,2) $|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}-\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)-\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}-\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}+\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)-\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}+\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Bpf}(2,2)+\frac{1}{3} \sqrt{\frac{5}{42}} \text{Bpf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)-\frac{9 \text{Apf}(2,2)}{5 \sqrt{14}}+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,2)+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{3}{35}} \text{Adf}(1,0)-\frac{\text{Adf}(3,0)}{2 \sqrt{105}}-\frac{\text{Adf}(3,2)}{3 \sqrt{14}}+\frac{5}{22} \sqrt{\frac{5}{21}} \text{Adf}(5,0)+\frac{5}{33} \sqrt{2} \text{Adf}(5,2)+\frac{5 \text{Adf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}}+\frac{5 \text{Bdf}(5,4)}{11 \sqrt{6}} }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Bff}(2,2)}{5 \sqrt{6}}-\frac{1}{11} \sqrt{10} \text{Bff}(4,2)-\frac{5}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)+\frac{25}{52} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)-\frac{2}{5} \sqrt{\frac{2}{3}} \text{Aff}(2,2)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Aff}(4,2)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{572} \sqrt{\frac{35}{3}} \text{Aff}(6,2)-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{25}{52} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$|$\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) }$|$ \frac{2}{3} \sqrt{\frac{2}{5}} \text{Bff}(2,2)+\frac{1}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)-\frac{40}{429} \sqrt{7} \text{Bff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ -\sqrt{\frac{6}{35}} \text{Bpf}(2,2)+\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4) }$|$\color{darkred}{ -\frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)+\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)-\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ -\frac{\text{Bff}(2,2)}{3 \sqrt{10}}-\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)+\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)+\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ \sqrt{\frac{6}{35}} \text{Bpf}(2,2)-\frac{\text{Bpf}(4,2)}{\sqrt{14}}+\frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\sqrt{\frac{3}{70}} \text{Apf}(2,2)-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}+\frac{\text{Adf}(3,0)}{6 \sqrt{7}}-\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{25 \text{Adf}(5,0)}{66 \sqrt{7}}-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2)+\frac{1}{11} \sqrt{\frac{5}{2}} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{11} \sqrt{\frac{5}{2}} \text{Bdf}(5,4)-\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Bff}(2,2)}{3 \sqrt{10}}+\frac{2}{11} \sqrt{\frac{2}{3}} \text{Bff}(4,2)+\frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)+\frac{5}{132} \sqrt{7} \text{Bff}(6,2)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4)+\frac{5}{52} \sqrt{\frac{35}{11}} \text{Bff}(6,6) $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{\text{Aff}(4,2)}{11 \sqrt{6}}-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{85 \sqrt{7} \text{Aff}(6,2)}{1716}+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{35}{11}} \text{Aff}(6,6) $|$ 0 $|$ \frac{\text{Bff}(2,2)}{\sqrt{6}}-\frac{1}{33} \sqrt{10} \text{Bff}(4,2)+\frac{35}{572} \sqrt{\frac{35}{3}} \text{Bff}(6,2)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Bff}(6,6) $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{7}{33} \sqrt{\frac{5}{2}} \text{Aff}(4,2)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)-\frac{5}{572} \sqrt{105} \text{Aff}(6,2)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4)-\frac{5}{52} \sqrt{\frac{21}{11}} \text{Aff}(6,6) $|$ 0 $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} \text{Apf}(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,2) $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}}+\frac{1}{11} \sqrt{10} \text{Adf}(5,4) }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,4) }$|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $|$ 0 $|$ 0 $|$ -\frac{2}{3} \sqrt{\frac{2}{5}} \text{Aff}(2,2)-\frac{1}{11} \sqrt{\frac{2}{3}} \text{Aff}(4,2)+\frac{40}{429} \sqrt{7} \text{Aff}(6,2) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $| |
| | |
| | |
| | ### |
| | |
| | ===== 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** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \text{Ea} & k=0\land m=0 \\ |
| | 0 & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{Ea, k == 0 && m == 0}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{0, 0, Ea} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \text{Ea} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ \text{s} $ ^ |
| | ^$ \text{s} $|$ \text{Ea} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ \text{s} $|$ 1 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_0_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{3} (\text{Ea}+\text{Ebx}+\text{Eby}) & k=0\land m=0 \\ |
| | 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| | \frac{5 (\text{Ebx}-\text{Eby}+2 i \text{Mb})}{2 \sqrt{6}} & k=2\land m=-2 \\ |
| | \frac{5}{6} (2 \text{Ea}-\text{Ebx}-\text{Eby}) & k=2\land m=0 \\ |
| | \frac{5 (\text{Ebx}-\text{Eby}-2 i \text{Mb})}{2 \sqrt{6}} & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Ea + Ebx + Eby)/3, k == 0 && m == 0}, {0, k != 2 || (m != -2 && m != 0 && m != 2)}, {(5*(Ebx - Eby + (2*I)*Mb))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(2*Ea - Ebx - Eby))/6, k == 2 && m == 0}}, (5*(Ebx - Eby - (2*I)*Mb))/(2*Sqrt[6])] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{0, 0, (1/3)*(Ea + Ebx + Eby)} , |
| | {2, 0, (5/6)*((2)*(Ea) + (-1)*(Ebx) + (-1)*(Eby))} , |
| | {2, 2, (5/2)*((1/(sqrt(6)))*(Ebx + (-1)*(Eby) + (-2*I)*(Mb)))} , |
| | {2,-2, (5/2)*((1/(sqrt(6)))*(Ebx + (-1)*(Eby) + (2*I)*(Mb)))} } |
| | |
| | </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{Ebx}+\text{Eby}}{2} $|$ 0 $|$ \frac{1}{2} (-\text{Ebx}+\text{Eby}-2 i \text{Mb}) $| |
| | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{1}{2} (-\text{Ebx}+\text{Eby}+2 i \text{Mb}) $|$ 0 $|$ \frac{\text{Ebx}+\text{Eby}}{2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ |
| | ^$ p_x $|$ \text{Ebx} $|$ \text{Mb} $|$ 0 $| |
| | ^$ p_y $|$ \text{Mb} $|$ \text{Eby} $|$ 0 $| |
| | ^$ p_z $|$ 0 $|$ 0 $|$ \text{Ea} $| |
| | |
| | |
| | ### |
| | |
| | </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** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ebx}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_1_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | |
| | ^ ^$$\text{Eby}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_1_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | |
| | ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_1_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{5} (\text{Eax2y2}+\text{Eaxy}+\text{Eaz2}+\text{Ebxz}+\text{Ebyz}) & k=0\land m=0 \\ |
| | 0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ |
| | \frac{\sqrt{3} \text{Ebxz}-\sqrt{3} \text{Ebyz}-4 \text{Maxy2yz2}-4 i \text{Maz2xy}+2 i \sqrt{3} \text{Mb}}{2 \sqrt{2}} & k=2\land m=-2 \\ |
| | \frac{1}{2} (-2 \text{Eax2y2}-2 \text{Eaxy}+2 \text{Eaz2}+\text{Ebxz}+\text{Ebyz}) & k=2\land m=0 \\ |
| | \frac{\sqrt{3} \text{Ebxz}-\sqrt{3} \text{Ebyz}-4 \text{Maxy2yz2}+4 i \text{Maz2xy}-2 i \sqrt{3} \text{Mb}}{2 \sqrt{2}} & k=2\land m=2 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eax2y2}-\text{Eaxy}+2 i \text{Max2y2xy}) & k=4\land m=-4 \\ |
| | \frac{3 \left(\text{Ebxz}-\text{Ebyz}+\sqrt{3} \text{Maxy2yz2}+i \sqrt{3} \text{Maz2xy}+2 i \text{Mb}\right)}{\sqrt{10}} & k=4\land m=-2 \\ |
| | \frac{3}{10} (\text{Eax2y2}+\text{Eaxy}+6 \text{Eaz2}-4 \text{Ebxz}-4 \text{Ebyz}) & k=4\land m=0 \\ |
| | \frac{3 \left(\text{Ebxz}-\text{Ebyz}+\sqrt{3} \text{Maxy2yz2}-i \sqrt{3} \text{Maz2xy}-2 i \text{Mb}\right)}{\sqrt{10}} & k=4\land m=2 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eax2y2}-\text{Eaxy}-2 i \text{Max2y2xy}) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eax2y2 + Eaxy + Eaz2 + Ebxz + Ebyz)/5, k == 0 && m == 0}, {0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {(Sqrt[3]*Ebxz - Sqrt[3]*Ebyz - 4*Maxy2yz2 - (4*I)*Maz2xy + (2*I)*Sqrt[3]*Mb)/(2*Sqrt[2]), k == 2 && m == -2}, {(-2*Eax2y2 - 2*Eaxy + 2*Eaz2 + Ebxz + Ebyz)/2, k == 2 && m == 0}, {(Sqrt[3]*Ebxz - Sqrt[3]*Ebyz - 4*Maxy2yz2 + (4*I)*Maz2xy - (2*I)*Sqrt[3]*Mb)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Eax2y2 - Eaxy + (2*I)*Max2y2xy))/2, k == 4 && m == -4}, {(3*(Ebxz - Ebyz + Sqrt[3]*Maxy2yz2 + I*Sqrt[3]*Maz2xy + (2*I)*Mb))/Sqrt[10], k == 4 && m == -2}, {(3*(Eax2y2 + Eaxy + 6*Eaz2 - 4*Ebxz - 4*Ebyz))/10, k == 4 && m == 0}, {(3*(Ebxz - Ebyz + Sqrt[3]*Maxy2yz2 - I*Sqrt[3]*Maz2xy - (2*I)*Mb))/Sqrt[10], k == 4 && m == 2}}, (3*Sqrt[7/10]*(Eax2y2 - Eaxy - (2*I)*Max2y2xy))/2] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{0, 0, (1/5)*(Eax2y2 + Eaxy + Eaz2 + Ebxz + Ebyz)} , |
| | {2, 0, (1/2)*((-2)*(Eax2y2) + (-2)*(Eaxy) + (2)*(Eaz2) + Ebxz + Ebyz)} , |
| | {2,-2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Ebxz) + (-1)*((sqrt(3))*(Ebyz)) + (-4)*(Maxy2yz2) + (-4*I)*(Maz2xy) + (2*I)*((sqrt(3))*(Mb))))} , |
| | {2, 2, (1/2)*((1/(sqrt(2)))*((sqrt(3))*(Ebxz) + (-1)*((sqrt(3))*(Ebyz)) + (-4)*(Maxy2yz2) + (4*I)*(Maz2xy) + (-2*I)*((sqrt(3))*(Mb))))} , |
| | {4, 0, (3/10)*(Eax2y2 + Eaxy + (6)*(Eaz2) + (-4)*(Ebxz) + (-4)*(Ebyz))} , |
| | {4, 2, (3)*((1/(sqrt(10)))*(Ebxz + (-1)*(Ebyz) + (sqrt(3))*(Maxy2yz2) + (-I)*((sqrt(3))*(Maz2xy)) + (-2*I)*(Mb)))} , |
| | {4,-2, (3)*((1/(sqrt(10)))*(Ebxz + (-1)*(Ebyz) + (sqrt(3))*(Maxy2yz2) + (I)*((sqrt(3))*(Maz2xy)) + (2*I)*(Mb)))} , |
| | {4, 4, (3/2)*((sqrt(7/10))*(Eax2y2 + (-1)*(Eaxy) + (-2*I)*(Max2y2xy)))} , |
| | {4,-4, (3/2)*((sqrt(7/10))*(Eax2y2 + (-1)*(Eaxy) + (2*I)*(Max2y2xy)))} } |
| | |
| | </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{Eax2y2}+\text{Eaxy}}{2} $|$ 0 $|$ \frac{\text{Maxy2yz2}+i \text{Maz2xy}}{\sqrt{2}} $|$ 0 $|$ \frac{1}{2} (\text{Eax2y2}-\text{Eaxy}+2 i \text{Max2y2xy}) $| |
| | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{\text{Ebxz}+\text{Ebyz}}{2} $|$ 0 $|$ \frac{1}{2} (-\text{Ebxz}+\text{Ebyz}-2 i \text{Mb}) $|$ 0 $| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Maxy2yz2}-i \text{Maz2xy}}{\sqrt{2}} $|$ 0 $|$ \text{Eaz2} $|$ 0 $|$ \frac{\text{Maxy2yz2}+i \text{Maz2xy}}{\sqrt{2}} $| |
| | ^$ {Y_{1}^{(2)}} $|$ 0 $|$ \frac{1}{2} (-\text{Ebxz}+\text{Ebyz}+2 i \text{Mb}) $|$ 0 $|$ \frac{\text{Ebxz}+\text{Ebyz}}{2} $|$ 0 $| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{1}{2} (\text{Eax2y2}-\text{Eaxy}-2 i \text{Max2y2xy}) $|$ 0 $|$ \frac{\text{Maxy2yz2}-i \text{Maz2xy}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Eax2y2}+\text{Eaxy}}{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{Eax2y2} $|$ \text{Maxy2yz2} $|$ 0 $|$ 0 $|$ \text{Max2y2xy} $| |
| | ^$ d_{3z^2-r^2} $|$ \text{Maxy2yz2} $|$ \text{Eaz2} $|$ 0 $|$ 0 $|$ \text{Maz2xy} $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Ebyz} $|$ \text{Mb} $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ \text{Mb} $|$ \text{Ebxz} $|$ 0 $| |
| | ^$ d_{\text{xy}} $|$ \text{Max2y2xy} $|$ \text{Maz2xy} $|$ 0 $|$ 0 $|$ \text{Eaxy} $| |
| | |
| | |
| | ### |
| | |
| | </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** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Eax2y2}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_2_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | |
| | ^ ^$$\text{Eaz2}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_2_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Ebyz}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_2_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | |
| | ^ ^$$\text{Ebxz}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_2_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | |
| | ^ ^$$\text{Eaxy}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_2_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for f orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{7} (\text{Eax3}+\text{Eaxy2z2}+\text{Eay3}+\text{Eayz2x2}+\text{Ebxyz}+\text{Ebz3}+\text{Ebzx2y2}) & k=0\land m=0 \\ |
| | 0 & (k\neq 6\land (((k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2))\land k\neq 4)\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4)))\lor (m\neq -6\land m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4\land m\neq 6) \\ |
| | \frac{5}{56} \left(2 \left(\sqrt{6} \text{Eax3}-\sqrt{6} \text{Eay3}+\sqrt{10} (\text{Max3xy2z2}+\text{May3yz2x2}-2 \text{Mbz3zx2y2})\right)-i \left(\sqrt{6} \text{Max3y3}+\sqrt{10} \text{Max3yz2x2}+5 \sqrt{6} \text{Maxy2z2yz2x2}-\sqrt{10} \text{May3xy2z2}+4 \sqrt{10} \text{Mbxyzz3}\right)\right) & k=2\land m=-2 \\ |
| | -\frac{5}{14} \left(\text{Eax3}+\text{Eay3}-2 \text{Ebz3}-\sqrt{15} \text{Max3xy2z2}+\sqrt{15} \text{May3yz2x2}\right) & k=2\land m=0 \\ |
| | \frac{5}{56} \left(2 \left(\sqrt{6} \text{Eax3}-\sqrt{6} \text{Eay3}+\sqrt{10} (\text{Max3xy2z2}+\text{May3yz2x2}-2 \text{Mbz3zx2y2})\right)+i \left(\sqrt{6} \text{Max3y3}+\sqrt{10} \text{Max3yz2x2}+5 \sqrt{6} \text{Maxy2z2yz2x2}-\sqrt{10} \text{May3xy2z2}+4 \sqrt{10} \text{Mbxyzz3}\right)\right) & k=2\land m=2 \\ |
| | \frac{3 \left(3 \sqrt{5} \text{Eax3}-3 \sqrt{5} \text{Eaxy2z2}+3 \sqrt{5} \text{Eay3}-3 \sqrt{5} \text{Eayz2x2}-4 \sqrt{5} \text{Ebxyz}+4 \sqrt{5} \text{Ebzx2y2}+2 \sqrt{3} \text{Max3xy2z2}-8 i \sqrt{3} \text{Max3yz2x2}-8 i \sqrt{3} \text{May3xy2z2}-2 \sqrt{3} \text{May3yz2x2}+8 i \sqrt{5} \text{Mbxyzzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=-4 \\ |
| | \frac{3}{56} \left(-3 \sqrt{10} \text{Eax3}+7 \sqrt{10} \text{Eaxy2z2}+3 \sqrt{10} \text{Eay3}-7 \sqrt{10} \text{Eayz2x2}+2 \sqrt{6} \text{Max3xy2z2}+4 i \left(3 \sqrt{10} \text{Max3y3}-2 \sqrt{6} \text{Max3yz2x2}+\sqrt{10} \text{Maxy2z2yz2x2}+2 \sqrt{6} \text{May3xy2z2}-\sqrt{6} \text{Mbxyzz3}\right)+2 \sqrt{6} \text{May3yz2x2}-4 \sqrt{6} \text{Mbz3zx2y2}\right) & k=4\land m=-2 \\ |
| | \frac{3}{56} \left(9 \text{Eax3}+7 \text{Eaxy2z2}+9 \text{Eay3}+7 \text{Eayz2x2}-28 \text{Ebxyz}+24 \text{Ebz3}-28 \text{Ebzx2y2}-2 \sqrt{15} \text{Max3xy2z2}+2 \sqrt{15} \text{May3yz2x2}\right) & k=4\land m=0 \\ |
| | \frac{3}{56} \left(-3 \sqrt{10} \text{Eax3}+7 \sqrt{10} \text{Eaxy2z2}+3 \sqrt{10} \text{Eay3}-7 \sqrt{10} \text{Eayz2x2}+2 \sqrt{6} \text{Max3xy2z2}-4 i \left(3 \sqrt{10} \text{Max3y3}-2 \sqrt{6} \text{Max3yz2x2}+\sqrt{10} \text{Maxy2z2yz2x2}+2 \sqrt{6} \text{May3xy2z2}-\sqrt{6} \text{Mbxyzz3}\right)+2 \sqrt{6} \text{May3yz2x2}-4 \sqrt{6} \text{Mbz3zx2y2}\right) & k=4\land m=2 \\ |
| | \frac{3 \left(3 \sqrt{5} \text{Eax3}-3 \sqrt{5} \text{Eaxy2z2}+3 \sqrt{5} \text{Eay3}-3 \sqrt{5} \text{Eayz2x2}-4 \sqrt{5} \text{Ebxyz}+4 \sqrt{5} \text{Ebzx2y2}+2 \sqrt{3} \text{Max3xy2z2}+8 i \sqrt{3} \text{Max3yz2x2}+8 i \sqrt{3} \text{May3xy2z2}-2 \sqrt{3} \text{May3yz2x2}-8 i \sqrt{5} \text{Mbxyzzx2y2}\right)}{8 \sqrt{14}} & k=4\land m=4 \\ |
| | \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eax3}+3 \sqrt{3} \text{Eaxy2z2}-5 \sqrt{3} \text{Eay3}-3 \sqrt{3} \text{Eayz2x2}-6 \sqrt{5} \text{Max3xy2z2}-10 i \sqrt{3} \text{Max3y3}-6 i \sqrt{5} \text{Max3yz2x2}+6 i \sqrt{3} \text{Maxy2z2yz2x2}+6 i \sqrt{5} \text{May3xy2z2}-6 \sqrt{5} \text{May3yz2x2}\right) & k=6\land m=-6 \\ |
| | -\frac{13 \left(15 \text{Eax3}-15 \text{Eaxy2z2}+15 \text{Eay3}-15 \text{Eayz2x2}+24 \text{Ebxyz}-24 \text{Ebzx2y2}+2 \sqrt{15} \text{Max3xy2z2}-8 i \sqrt{15} \text{Max3yz2x2}-8 i \sqrt{15} \text{May3xy2z2}-2 \sqrt{15} \text{May3yz2x2}-48 i \text{Mbxyzzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=-4 \\ |
| | \frac{13 \left(5 \sqrt{15} \text{Eax3}+3 \sqrt{15} \text{Eaxy2z2}-5 \sqrt{15} \text{Eay3}-3 \sqrt{15} \text{Eayz2x2}+34 \text{Max3xy2z2}+2 i \sqrt{15} \text{Max3y3}-26 i \text{Max3yz2x2}-14 i \sqrt{15} \text{Maxy2z2yz2x2}+26 i \text{May3xy2z2}+34 \text{May3yz2x2}+64 i \text{Mbxyzz3}+64 \text{Mbz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=-2 \\ |
| | -\frac{13}{560} \left(25 \text{Eax3}+39 \text{Eaxy2z2}+25 \text{Eay3}+39 \text{Eayz2x2}-24 \text{Ebxyz}-80 \text{Ebz3}-24 \text{Ebzx2y2}+14 \sqrt{15} \text{Max3xy2z2}-14 \sqrt{15} \text{May3yz2x2}\right) & k=6\land m=0 \\ |
| | \frac{13 \left(5 \sqrt{15} \text{Eax3}+3 \sqrt{15} \text{Eaxy2z2}-5 \sqrt{15} \text{Eay3}-3 \sqrt{15} \text{Eayz2x2}+34 \text{Max3xy2z2}-2 i \sqrt{15} \text{Max3y3}+26 i \text{Max3yz2x2}+14 i \sqrt{15} \text{Maxy2z2yz2x2}-26 i \text{May3xy2z2}+34 \text{May3yz2x2}-64 i \text{Mbxyzz3}+64 \text{Mbz3zx2y2}\right)}{160 \sqrt{7}} & k=6\land m=2 \\ |
| | -\frac{13 \left(15 \text{Eax3}-15 \text{Eaxy2z2}+15 \text{Eay3}-15 \text{Eayz2x2}+24 \text{Ebxyz}-24 \text{Ebzx2y2}+2 \sqrt{15} \text{Max3xy2z2}+8 i \sqrt{15} \text{Max3yz2x2}+8 i \sqrt{15} \text{May3xy2z2}-2 \sqrt{15} \text{May3yz2x2}+48 i \text{Mbxyzzx2y2}\right)}{80 \sqrt{14}} & k=6\land m=4 \\ |
| | \frac{13}{160} \sqrt{\frac{11}{7}} \left(5 \sqrt{3} \text{Eax3}+3 \sqrt{3} \text{Eaxy2z2}-5 \sqrt{3} \text{Eay3}-3 \sqrt{3} \text{Eayz2x2}-6 \sqrt{5} \text{Max3xy2z2}+10 i \sqrt{3} \text{Max3y3}+6 i \sqrt{5} \text{Max3yz2x2}-6 i \sqrt{3} \text{Maxy2z2yz2x2}-6 i \sqrt{5} \text{May3xy2z2}-6 \sqrt{5} \text{May3yz2x2}\right) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Eax3 + Eaxy2z2 + Eay3 + Eayz2x2 + Ebxyz + Ebz3 + Ebzx2y2)/7, k == 0 && m == 0}, {0, (k != 6 && (((k != 2 || (m != -2 && m != 0 && m != 2)) && k != 4) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4))) || (m != -6 && m != -4 && m != -2 && m != 0 && m != 2 && m != 4 && m != 6)}, {(5*((-I)*(Sqrt[6]*Max3y3 + Sqrt[10]*Max3yz2x2 + 5*Sqrt[6]*Maxy2z2yz2x2 - Sqrt[10]*May3xy2z2 + 4*Sqrt[10]*Mbxyzz3) + 2*(Sqrt[6]*Eax3 - Sqrt[6]*Eay3 + Sqrt[10]*(Max3xy2z2 + May3yz2x2 - 2*Mbz3zx2y2))))/56, k == 2 && m == -2}, {(-5*(Eax3 + Eay3 - 2*Ebz3 - Sqrt[15]*Max3xy2z2 + Sqrt[15]*May3yz2x2))/14, k == 2 && m == 0}, {(5*(I*(Sqrt[6]*Max3y3 + Sqrt[10]*Max3yz2x2 + 5*Sqrt[6]*Maxy2z2yz2x2 - Sqrt[10]*May3xy2z2 + 4*Sqrt[10]*Mbxyzz3) + 2*(Sqrt[6]*Eax3 - Sqrt[6]*Eay3 + Sqrt[10]*(Max3xy2z2 + May3yz2x2 - 2*Mbz3zx2y2))))/56, k == 2 && m == 2}, {(3*(3*Sqrt[5]*Eax3 - 3*Sqrt[5]*Eaxy2z2 + 3*Sqrt[5]*Eay3 - 3*Sqrt[5]*Eayz2x2 - 4*Sqrt[5]*Ebxyz + 4*Sqrt[5]*Ebzx2y2 + 2*Sqrt[3]*Max3xy2z2 - (8*I)*Sqrt[3]*Max3yz2x2 - (8*I)*Sqrt[3]*May3xy2z2 - 2*Sqrt[3]*May3yz2x2 + (8*I)*Sqrt[5]*Mbxyzzx2y2))/(8*Sqrt[14]), k == 4 && m == -4}, {(3*(-3*Sqrt[10]*Eax3 + 7*Sqrt[10]*Eaxy2z2 + 3*Sqrt[10]*Eay3 - 7*Sqrt[10]*Eayz2x2 + 2*Sqrt[6]*Max3xy2z2 + 2*Sqrt[6]*May3yz2x2 + (4*I)*(3*Sqrt[10]*Max3y3 - 2*Sqrt[6]*Max3yz2x2 + Sqrt[10]*Maxy2z2yz2x2 + 2*Sqrt[6]*May3xy2z2 - Sqrt[6]*Mbxyzz3) - 4*Sqrt[6]*Mbz3zx2y2))/56, k == 4 && m == -2}, {(3*(9*Eax3 + 7*Eaxy2z2 + 9*Eay3 + 7*Eayz2x2 - 28*Ebxyz + 24*Ebz3 - 28*Ebzx2y2 - 2*Sqrt[15]*Max3xy2z2 + 2*Sqrt[15]*May3yz2x2))/56, k == 4 && m == 0}, {(3*(-3*Sqrt[10]*Eax3 + 7*Sqrt[10]*Eaxy2z2 + 3*Sqrt[10]*Eay3 - 7*Sqrt[10]*Eayz2x2 + 2*Sqrt[6]*Max3xy2z2 + 2*Sqrt[6]*May3yz2x2 - (4*I)*(3*Sqrt[10]*Max3y3 - 2*Sqrt[6]*Max3yz2x2 + Sqrt[10]*Maxy2z2yz2x2 + 2*Sqrt[6]*May3xy2z2 - Sqrt[6]*Mbxyzz3) - 4*Sqrt[6]*Mbz3zx2y2))/56, k == 4 && m == 2}, {(3*(3*Sqrt[5]*Eax3 - 3*Sqrt[5]*Eaxy2z2 + 3*Sqrt[5]*Eay3 - 3*Sqrt[5]*Eayz2x2 - 4*Sqrt[5]*Ebxyz + 4*Sqrt[5]*Ebzx2y2 + 2*Sqrt[3]*Max3xy2z2 + (8*I)*Sqrt[3]*Max3yz2x2 + (8*I)*Sqrt[3]*May3xy2z2 - 2*Sqrt[3]*May3yz2x2 - (8*I)*Sqrt[5]*Mbxyzzx2y2))/(8*Sqrt[14]), k == 4 && m == 4}, {(13*Sqrt[11/7]*(5*Sqrt[3]*Eax3 + 3*Sqrt[3]*Eaxy2z2 - 5*Sqrt[3]*Eay3 - 3*Sqrt[3]*Eayz2x2 - 6*Sqrt[5]*Max3xy2z2 - (10*I)*Sqrt[3]*Max3y3 - (6*I)*Sqrt[5]*Max3yz2x2 + (6*I)*Sqrt[3]*Maxy2z2yz2x2 + (6*I)*Sqrt[5]*May3xy2z2 - 6*Sqrt[5]*May3yz2x2))/160, k == 6 && m == -6}, {(-13*(15*Eax3 - 15*Eaxy2z2 + 15*Eay3 - 15*Eayz2x2 + 24*Ebxyz - 24*Ebzx2y2 + 2*Sqrt[15]*Max3xy2z2 - (8*I)*Sqrt[15]*Max3yz2x2 - (8*I)*Sqrt[15]*May3xy2z2 - 2*Sqrt[15]*May3yz2x2 - (48*I)*Mbxyzzx2y2))/(80*Sqrt[14]), k == 6 && m == -4}, {(13*(5*Sqrt[15]*Eax3 + 3*Sqrt[15]*Eaxy2z2 - 5*Sqrt[15]*Eay3 - 3*Sqrt[15]*Eayz2x2 + 34*Max3xy2z2 + (2*I)*Sqrt[15]*Max3y3 - (26*I)*Max3yz2x2 - (14*I)*Sqrt[15]*Maxy2z2yz2x2 + (26*I)*May3xy2z2 + 34*May3yz2x2 + (64*I)*Mbxyzz3 + 64*Mbz3zx2y2))/(160*Sqrt[7]), k == 6 && m == -2}, {(-13*(25*Eax3 + 39*Eaxy2z2 + 25*Eay3 + 39*Eayz2x2 - 24*Ebxyz - 80*Ebz3 - 24*Ebzx2y2 + 14*Sqrt[15]*Max3xy2z2 - 14*Sqrt[15]*May3yz2x2))/560, k == 6 && m == 0}, {(13*(5*Sqrt[15]*Eax3 + 3*Sqrt[15]*Eaxy2z2 - 5*Sqrt[15]*Eay3 - 3*Sqrt[15]*Eayz2x2 + 34*Max3xy2z2 - (2*I)*Sqrt[15]*Max3y3 + (26*I)*Max3yz2x2 + (14*I)*Sqrt[15]*Maxy2z2yz2x2 - (26*I)*May3xy2z2 + 34*May3yz2x2 - (64*I)*Mbxyzz3 + 64*Mbz3zx2y2))/(160*Sqrt[7]), k == 6 && m == 2}, {(-13*(15*Eax3 - 15*Eaxy2z2 + 15*Eay3 - 15*Eayz2x2 + 24*Ebxyz - 24*Ebzx2y2 + 2*Sqrt[15]*Max3xy2z2 + (8*I)*Sqrt[15]*Max3yz2x2 + (8*I)*Sqrt[15]*May3xy2z2 - 2*Sqrt[15]*May3yz2x2 + (48*I)*Mbxyzzx2y2))/(80*Sqrt[14]), k == 6 && m == 4}}, (13*Sqrt[11/7]*(5*Sqrt[3]*Eax3 + 3*Sqrt[3]*Eaxy2z2 - 5*Sqrt[3]*Eay3 - 3*Sqrt[3]*Eayz2x2 - 6*Sqrt[5]*Max3xy2z2 + (10*I)*Sqrt[3]*Max3y3 + (6*I)*Sqrt[5]*Max3yz2x2 - (6*I)*Sqrt[3]*Maxy2z2yz2x2 - (6*I)*Sqrt[5]*May3xy2z2 - 6*Sqrt[5]*May3yz2x2))/160] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{0, 0, (1/7)*(Eax3 + Eaxy2z2 + Eay3 + Eayz2x2 + Ebxyz + Ebz3 + Ebzx2y2)} , |
| | {2, 0, (-5/14)*(Eax3 + Eay3 + (-2)*(Ebz3) + (-1)*((sqrt(15))*(Max3xy2z2)) + (sqrt(15))*(May3yz2x2))} , |
| | {2,-2, (5/56)*((-I)*((sqrt(6))*(Max3y3) + (sqrt(10))*(Max3yz2x2) + (5)*((sqrt(6))*(Maxy2z2yz2x2)) + (-1)*((sqrt(10))*(May3xy2z2)) + (4)*((sqrt(10))*(Mbxyzz3))) + (2)*((sqrt(6))*(Eax3) + (-1)*((sqrt(6))*(Eay3)) + (sqrt(10))*(Max3xy2z2 + May3yz2x2 + (-2)*(Mbz3zx2y2))))} , |
| | {2, 2, (5/56)*((I)*((sqrt(6))*(Max3y3) + (sqrt(10))*(Max3yz2x2) + (5)*((sqrt(6))*(Maxy2z2yz2x2)) + (-1)*((sqrt(10))*(May3xy2z2)) + (4)*((sqrt(10))*(Mbxyzz3))) + (2)*((sqrt(6))*(Eax3) + (-1)*((sqrt(6))*(Eay3)) + (sqrt(10))*(Max3xy2z2 + May3yz2x2 + (-2)*(Mbz3zx2y2))))} , |
| | {4, 0, (3/56)*((9)*(Eax3) + (7)*(Eaxy2z2) + (9)*(Eay3) + (7)*(Eayz2x2) + (-28)*(Ebxyz) + (24)*(Ebz3) + (-28)*(Ebzx2y2) + (-2)*((sqrt(15))*(Max3xy2z2)) + (2)*((sqrt(15))*(May3yz2x2)))} , |
| | {4, 2, (3/56)*((-3)*((sqrt(10))*(Eax3)) + (7)*((sqrt(10))*(Eaxy2z2)) + (3)*((sqrt(10))*(Eay3)) + (-7)*((sqrt(10))*(Eayz2x2)) + (2)*((sqrt(6))*(Max3xy2z2)) + (2)*((sqrt(6))*(May3yz2x2)) + (-4*I)*((3)*((sqrt(10))*(Max3y3)) + (-2)*((sqrt(6))*(Max3yz2x2)) + (sqrt(10))*(Maxy2z2yz2x2) + (2)*((sqrt(6))*(May3xy2z2)) + (-1)*((sqrt(6))*(Mbxyzz3))) + (-4)*((sqrt(6))*(Mbz3zx2y2)))} , |
| | {4,-2, (3/56)*((-3)*((sqrt(10))*(Eax3)) + (7)*((sqrt(10))*(Eaxy2z2)) + (3)*((sqrt(10))*(Eay3)) + (-7)*((sqrt(10))*(Eayz2x2)) + (2)*((sqrt(6))*(Max3xy2z2)) + (2)*((sqrt(6))*(May3yz2x2)) + (4*I)*((3)*((sqrt(10))*(Max3y3)) + (-2)*((sqrt(6))*(Max3yz2x2)) + (sqrt(10))*(Maxy2z2yz2x2) + (2)*((sqrt(6))*(May3xy2z2)) + (-1)*((sqrt(6))*(Mbxyzz3))) + (-4)*((sqrt(6))*(Mbz3zx2y2)))} , |
| | {4,-4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eax3)) + (-3)*((sqrt(5))*(Eaxy2z2)) + (3)*((sqrt(5))*(Eay3)) + (-3)*((sqrt(5))*(Eayz2x2)) + (-4)*((sqrt(5))*(Ebxyz)) + (4)*((sqrt(5))*(Ebzx2y2)) + (2)*((sqrt(3))*(Max3xy2z2)) + (-8*I)*((sqrt(3))*(Max3yz2x2)) + (-8*I)*((sqrt(3))*(May3xy2z2)) + (-2)*((sqrt(3))*(May3yz2x2)) + (8*I)*((sqrt(5))*(Mbxyzzx2y2))))} , |
| | {4, 4, (3/8)*((1/(sqrt(14)))*((3)*((sqrt(5))*(Eax3)) + (-3)*((sqrt(5))*(Eaxy2z2)) + (3)*((sqrt(5))*(Eay3)) + (-3)*((sqrt(5))*(Eayz2x2)) + (-4)*((sqrt(5))*(Ebxyz)) + (4)*((sqrt(5))*(Ebzx2y2)) + (2)*((sqrt(3))*(Max3xy2z2)) + (8*I)*((sqrt(3))*(Max3yz2x2)) + (8*I)*((sqrt(3))*(May3xy2z2)) + (-2)*((sqrt(3))*(May3yz2x2)) + (-8*I)*((sqrt(5))*(Mbxyzzx2y2))))} , |
| | {6, 0, (-13/560)*((25)*(Eax3) + (39)*(Eaxy2z2) + (25)*(Eay3) + (39)*(Eayz2x2) + (-24)*(Ebxyz) + (-80)*(Ebz3) + (-24)*(Ebzx2y2) + (14)*((sqrt(15))*(Max3xy2z2)) + (-14)*((sqrt(15))*(May3yz2x2)))} , |
| | {6, 2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eax3)) + (3)*((sqrt(15))*(Eaxy2z2)) + (-5)*((sqrt(15))*(Eay3)) + (-3)*((sqrt(15))*(Eayz2x2)) + (34)*(Max3xy2z2) + (-2*I)*((sqrt(15))*(Max3y3)) + (26*I)*(Max3yz2x2) + (14*I)*((sqrt(15))*(Maxy2z2yz2x2)) + (-26*I)*(May3xy2z2) + (34)*(May3yz2x2) + (-64*I)*(Mbxyzz3) + (64)*(Mbz3zx2y2)))} , |
| | {6,-2, (13/160)*((1/(sqrt(7)))*((5)*((sqrt(15))*(Eax3)) + (3)*((sqrt(15))*(Eaxy2z2)) + (-5)*((sqrt(15))*(Eay3)) + (-3)*((sqrt(15))*(Eayz2x2)) + (34)*(Max3xy2z2) + (2*I)*((sqrt(15))*(Max3y3)) + (-26*I)*(Max3yz2x2) + (-14*I)*((sqrt(15))*(Maxy2z2yz2x2)) + (26*I)*(May3xy2z2) + (34)*(May3yz2x2) + (64*I)*(Mbxyzz3) + (64)*(Mbz3zx2y2)))} , |
| | {6,-4, (-13/80)*((1/(sqrt(14)))*((15)*(Eax3) + (-15)*(Eaxy2z2) + (15)*(Eay3) + (-15)*(Eayz2x2) + (24)*(Ebxyz) + (-24)*(Ebzx2y2) + (2)*((sqrt(15))*(Max3xy2z2)) + (-8*I)*((sqrt(15))*(Max3yz2x2)) + (-8*I)*((sqrt(15))*(May3xy2z2)) + (-2)*((sqrt(15))*(May3yz2x2)) + (-48*I)*(Mbxyzzx2y2)))} , |
| | {6, 4, (-13/80)*((1/(sqrt(14)))*((15)*(Eax3) + (-15)*(Eaxy2z2) + (15)*(Eay3) + (-15)*(Eayz2x2) + (24)*(Ebxyz) + (-24)*(Ebzx2y2) + (2)*((sqrt(15))*(Max3xy2z2)) + (8*I)*((sqrt(15))*(Max3yz2x2)) + (8*I)*((sqrt(15))*(May3xy2z2)) + (-2)*((sqrt(15))*(May3yz2x2)) + (48*I)*(Mbxyzzx2y2)))} , |
| | {6,-6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eax3)) + (3)*((sqrt(3))*(Eaxy2z2)) + (-5)*((sqrt(3))*(Eay3)) + (-3)*((sqrt(3))*(Eayz2x2)) + (-6)*((sqrt(5))*(Max3xy2z2)) + (-10*I)*((sqrt(3))*(Max3y3)) + (-6*I)*((sqrt(5))*(Max3yz2x2)) + (6*I)*((sqrt(3))*(Maxy2z2yz2x2)) + (6*I)*((sqrt(5))*(May3xy2z2)) + (-6)*((sqrt(5))*(May3yz2x2))))} , |
| | {6, 6, (13/160)*((sqrt(11/7))*((5)*((sqrt(3))*(Eax3)) + (3)*((sqrt(3))*(Eaxy2z2)) + (-5)*((sqrt(3))*(Eay3)) + (-3)*((sqrt(3))*(Eayz2x2)) + (-6)*((sqrt(5))*(Max3xy2z2)) + (10*I)*((sqrt(3))*(Max3y3)) + (6*I)*((sqrt(5))*(Max3yz2x2)) + (-6*I)*((sqrt(3))*(Maxy2z2yz2x2)) + (-6*I)*((sqrt(5))*(May3xy2z2)) + (-6)*((sqrt(5))*(May3yz2x2))))} } |
| | |
| | </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{Eax3}+3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \sqrt{15} (\text{May3yz2x2}-\text{Max3xy2z2})\right) $|$ 0 $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}+2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}+i \text{May3yz2x2}\right)\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}-4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) $|$ 0 $|$ \frac{1}{16} \left(-5 \text{Eax3}-3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \left(\sqrt{15} \text{Max3xy2z2}+5 i \text{Max3y3}+i \sqrt{15} \text{Max3yz2x2}-3 i \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3yz2x2}-i \text{May3xy2z2})\right)\right) $| |
| | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Ebxyz}+\text{Ebzx2y2}}{2} $|$ 0 $|$ \frac{\text{Mbz3zx2y2}+i \text{Mbxyzz3}}{\sqrt{2}} $|$ 0 $|$ \frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}+2 i \text{Mbxyzzx2y2}) $|$ 0 $| |
| | ^$ {Y_{-1}^{(3)}} $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}-2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}-i \text{May3yz2x2}\right)\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eax3}+5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}+2 \sqrt{15} (\text{Max3xy2z2}-\text{May3yz2x2})\right) $|$ 0 $|$ \frac{1}{16} \left(-3 \text{Eax3}-5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}-2 \left(\sqrt{15} \text{Max3xy2z2}+3 i \text{Max3y3}-i \sqrt{15} \text{Max3yz2x2}-5 i \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3yz2x2}+i \text{May3xy2z2})\right)\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}-4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) $| |
| | ^$ {Y_{0}^{(3)}} $|$ 0 $|$ \frac{\text{Mbz3zx2y2}-i \text{Mbxyzz3}}{\sqrt{2}} $|$ 0 $|$ \text{Ebz3} $|$ 0 $|$ \frac{\text{Mbz3zx2y2}+i \text{Mbxyzz3}}{\sqrt{2}} $|$ 0 $| |
| | ^$ {Y_{1}^{(3)}} $|$ \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}+4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) $|$ 0 $|$ \frac{1}{16} \left(-3 \text{Eax3}-5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}-2 \sqrt{15} \text{Max3xy2z2}+2 i \left(3 \text{Max3y3}-\sqrt{15} \text{Max3yz2x2}-5 \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3xy2z2}+i \text{May3yz2x2})\right)\right) $|$ 0 $|$ \frac{1}{16} \left(3 \text{Eax3}+5 \text{Eaxy2z2}+3 \text{Eay3}+5 \text{Eayz2x2}+2 \sqrt{15} (\text{Max3xy2z2}-\text{May3yz2x2})\right) $|$ 0 $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}+2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}+i \text{May3yz2x2}\right)\right) $| |
| | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{1}{2} (-\text{Ebxyz}+\text{Ebzx2y2}-2 i \text{Mbxyzzx2y2}) $|$ 0 $|$ \frac{\text{Mbz3zx2y2}-i \text{Mbxyzz3}}{\sqrt{2}} $|$ 0 $|$ \frac{\text{Ebxyz}+\text{Ebzx2y2}}{2} $|$ 0 $| |
| | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{16} \left(-5 \text{Eax3}-3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \left(\sqrt{15} \text{Max3xy2z2}-5 i \text{Max3y3}-i \sqrt{15} \text{Max3yz2x2}+3 i \text{Maxy2z2yz2x2}+\sqrt{15} (\text{May3yz2x2}+i \text{May3xy2z2})\right)\right) $|$ 0 $|$ \frac{1}{16} \left(\sqrt{15} \text{Eax3}-\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}+2 (\text{Max3xy2z2}+4 i (\text{Max3yz2x2}+\text{May3xy2z2})-\text{May3yz2x2})\right) $|$ 0 $|$ \frac{1}{16} \left(-\sqrt{15} \text{Eax3}+\sqrt{15} \text{Eaxy2z2}+\sqrt{15} \text{Eay3}-\sqrt{15} \text{Eayz2x2}-2 \text{Max3xy2z2}-2 i \left(\sqrt{15} \text{Max3y3}-\text{Max3yz2x2}+\sqrt{15} \text{Maxy2z2yz2x2}+\text{May3xy2z2}-i \text{May3yz2x2}\right)\right) $|$ 0 $|$ \frac{1}{16} \left(5 \text{Eax3}+3 \text{Eaxy2z2}+5 \text{Eay3}+3 \text{Eayz2x2}+2 \sqrt{15} (\text{May3yz2x2}-\text{Max3xy2z2})\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{Ebxyz} $|$ 0 $|$ 0 $|$ \text{Mbxyzz3} $|$ 0 $|$ 0 $|$ \text{Mbxyzzx2y2} $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Eax3} $|$ \text{Max3y3} $|$ 0 $|$ \text{Max3xy2z2} $|$ \text{Max3yz2x2} $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ \text{Max3y3} $|$ \text{Eay3} $|$ 0 $|$ \text{May3xy2z2} $|$ \text{May3yz2x2} $|$ 0 $| |
| | ^$ f_{z\left(5z^2-r^2\right)} $|$ \text{Mbxyzz3} $|$ 0 $|$ 0 $|$ \text{Ebz3} $|$ 0 $|$ 0 $|$ \text{Mbz3zx2y2} $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \text{Max3xy2z2} $|$ \text{May3xy2z2} $|$ 0 $|$ \text{Eaxy2z2} $|$ \text{Maxy2z2yz2x2} $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ \text{Max3yz2x2} $|$ \text{May3yz2x2} $|$ 0 $|$ \text{Maxy2z2yz2x2} $|$ \text{Eayz2x2} $|$ 0 $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$ \text{Mbxyzzx2y2} $|$ 0 $|$ 0 $|$ \text{Mbz3zx2y2} $|$ 0 $|$ 0 $|$ \text{Ebzx2y2} $| |
| | |
| | |
| | ### |
| | |
| | </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** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Ebxyz}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_1.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | |
| | ^ ^$$\text{Eax3}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_2.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\text{Eay3}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_3.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\text{Ebz3}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_4.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | |
| | ^ ^$$\text{Eaxy2z2}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_5.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Eayz2x2}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_6.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Ebzx2y2}$$ | {{:physics_chemistry:pointgroup:c2_z_orb_3_7.png?150}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ===== Coupling between two shells ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| | |
| | ==== Potential for s-p orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 1\lor m\neq 0 \\ |
| | A(1,0) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 1 || m != 0}}, A[1, 0]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{1, 0, A(1,0)} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{A(1,0)}{\sqrt{3}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ |
| | ^$ \text{s} $|$ 0 $|$ 0 $|$ \frac{A(1,0)}{\sqrt{3}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for s-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 2 || (m != -2 && m != 0 && m != 2)}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}}, A[2, 2] + I*B[2, 2]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \frac{A(2,2)+i B(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,2)-i B(2,2)}{\sqrt{5}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ \text{s} $|$ \sqrt{\frac{2}{5}} A(2,2) $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} B(2,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for s-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & k\neq 3\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| | A(3,2)-i B(3,2) & k=3\land m=-2 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,2)+i B(3,2) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 0 && m != 2)}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}}, A[3, 2] + I*B[3, 2]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{3, 0, A(3,0)} , |
| | {3,-2, A(3,2) + (-I)*(B(3,2))} , |
| | {3, 2, A(3,2) + (I)*(B(3,2))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{A(3,2)+i B(3,2)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,2)-i B(3,2)}{\sqrt{7}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ \text{s} $|$ -\sqrt{\frac{2}{7}} B(3,2) $|$ 0 $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{7}} A(3,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 3\land (k\neq 1\lor m\neq 0))\lor (m\neq -2\land m\neq 0\land m\neq 2) \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(3,2)-i B(3,2) & k=3\land m=-2 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,2)+i B(3,2) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || m != 0)) || (m != -2 && m != 0 && m != 2)}, {A[1, 0], k == 1 && m == 0}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}}, A[3, 2] + I*B[3, 2]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{1, 0, A(1,0)} , |
| | {3, 0, A(3,0)} , |
| | {3,-2, A(3,2) + (-I)*(B(3,2))} , |
| | {3, 2, A(3,2) + (I)*(B(3,2))} } |
| | |
| | </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_{-1}^{(1)}} $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ -\frac{1}{7} \sqrt{6} (A(3,2)-i B(3,2)) $|$ 0 $| |
| | ^$ {Y_{0}^{(1)}} $|$ \frac{1}{7} \sqrt{3} (A(3,2)+i B(3,2)) $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ \frac{1}{7} \sqrt{3} (A(3,2)-i B(3,2)) $| |
| | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ -\frac{1}{7} \sqrt{6} (A(3,2)+i B(3,2)) $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ p_x $|$ 0 $|$ 0 $|$ -\frac{1}{7} \sqrt{6} B(3,2) $|$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)+5 \sqrt{6} A(3,2)\right) $|$ 0 $| |
| | ^$ p_y $|$ 0 $|$ 0 $|$ \frac{1}{35} \left(7 \sqrt{5} A(1,0)-3 \sqrt{5} A(3,0)-5 \sqrt{6} A(3,2)\right) $|$ -\frac{1}{7} \sqrt{6} B(3,2) $|$ 0 $| |
| | ^$ p_z $|$ \frac{1}{7} \sqrt{6} A(3,2) $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $|$ -\frac{1}{7} \sqrt{6} B(3,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 4\land (k\neq 2\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}}, A[4, 4] + I*B[4, 4]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {4, 0, A(4,0)} , |
| | {4,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{3 (A(2,2)+i B(2,2))}{\sqrt{35}}-\frac{A(4,2)+i B(4,2)}{3 \sqrt{21}} $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i B(4,2)) $|$ 0 $|$ -\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \sqrt{\frac{3}{35}} (A(2,2)+i B(2,2))+\frac{2 (A(4,2)+i B(4,2))}{3 \sqrt{7}} $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} (A(2,2)-i B(2,2))+\frac{2 (A(4,2)-i B(4,2))}{3 \sqrt{7}} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$ -\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i B(4,2)) $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{3 (A(2,2)-i B(2,2))}{\sqrt{35}}-\frac{A(4,2)-i B(4,2)}{3 \sqrt{21}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ p_x $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)+7 B(4,4)\right)\right) $|$ 0 $|$ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ \sqrt{\frac{6}{35}} B(2,2)-\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_y $|$ 0 $|$ \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)-7 B(4,4)\right)\right) $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ 0 $|$ -\sqrt{\frac{6}{35}} B(2,2)+\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} $|$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ 0 $| |
| | ^$ p_z $|$ -\sqrt{\frac{6}{35}} B(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} B(4,2) $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | 0 & (k\neq 5\land (((k\neq 1\lor m\neq 0)\land k\neq 3)\lor (m\neq -2\land m\neq 0\land m\neq 2)))\lor (m\neq -4\land m\neq -2\land m\neq 0\land m\neq 2\land m\neq 4) \\ |
| | A(1,0) & k=1\land m=0 \\ |
| | A(3,2)-i B(3,2) & k=3\land m=-2 \\ |
| | A(3,0) & k=3\land m=0 \\ |
| | A(3,2)+i B(3,2) & k=3\land m=2 \\ |
| | A(5,4)-i B(5,4) & k=5\land m=-4 \\ |
| | A(5,2)-i B(5,2) & k=5\land m=-2 \\ |
| | A(5,0) & k=5\land m=0 \\ |
| | A(5,2)+i B(5,2) & k=5\land m=2 \\ |
| | A(5,4)+i B(5,4) & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{0, (k != 5 && (((k != 1 || m != 0) && k != 3) || (m != -2 && m != 0 && m != 2))) || (m != -4 && m != -2 && m != 0 && m != 2 && m != 4)}, {A[1, 0], k == 1 && m == 0}, {A[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 0], k == 3 && m == 0}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[5, 4] - I*B[5, 4], k == 5 && m == -4}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {A[5, 0], k == 5 && m == 0}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}}, A[5, 4] + I*B[5, 4]] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_C2_Z.Quanty> |
| | |
| | Akm = {{1, 0, A(1,0)} , |
| | {3, 0, A(3,0)} , |
| | {3,-2, A(3,2) + (-I)*(B(3,2))} , |
| | {3, 2, A(3,2) + (I)*(B(3,2))} , |
| | {5, 0, A(5,0)} , |
| | {5,-2, A(5,2) + (-I)*(B(5,2))} , |
| | {5, 2, A(5,2) + (I)*(B(5,2))} , |
| | {5,-4, A(5,4) + (-I)*(B(5,4))} , |
| | {5, 4, A(5,4) + (I)*(B(5,4))} } |
| | |
| | </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_{-2}^{(2)}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|$ \frac{5}{33} (A(5,2)-i B(5,2))-\frac{2 (A(3,2)-i B(3,2))}{3 \sqrt{7}} $|$ 0 $|$ \frac{1}{11} \sqrt{10} (A(5,4)-i B(5,4)) $|$ 0 $| |
| | ^$ {Y_{-1}^{(2)}} $|$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,2)+i B(3,2))-\frac{1}{33} \sqrt{5} (A(5,2)+i B(5,2)) $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ -\frac{A(3,2)-i B(3,2)}{\sqrt{21}}-\frac{5 (A(5,2)-i B(5,2))}{11 \sqrt{3}} $|$ 0 $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} (A(5,4)-i B(5,4)) $| |
| | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ \frac{1}{11} \sqrt{5} (A(5,2)+i B(5,2)) $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ \frac{1}{11} \sqrt{5} (A(5,2)-i B(5,2)) $|$ 0 $| |
| | ^$ {Y_{1}^{(2)}} $|$ -\frac{2}{11} \sqrt{\frac{5}{3}} (A(5,4)+i B(5,4)) $|$ 0 $|$ -\frac{A(3,2)+i B(3,2)}{\sqrt{21}}-\frac{5 (A(5,2)+i B(5,2))}{11 \sqrt{3}} $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,2)-i B(3,2))-\frac{1}{33} \sqrt{5} (A(5,2)-i B(5,2)) $| |
| | ^$ {Y_{2}^{(2)}} $|$ 0 $|$ \frac{1}{11} \sqrt{10} (A(5,4)+i B(5,4)) $|$ 0 $|$ \frac{5}{33} (A(5,2)+i B(5,2))-\frac{2 (A(3,2)+i B(3,2))}{3 \sqrt{7}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ d_{x^2-y^2} $|$ -\frac{1}{11} \sqrt{10} B(5,4) $|$ 0 $|$ 0 $|$ \frac{5}{33} \sqrt{2} A(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} A(3,2) $|$ 0 $|$ 0 $|$ \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)+21 \sqrt{10} A(5,4)\right) $| |
| | ^$ d_{3z^2-r^2} $|$ -\frac{1}{11} \sqrt{10} B(5,2) $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{10} A(5,2) $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ \frac{44 \sqrt{7} B(3,2)+35 \left(B(5,2)-\sqrt{3} B(5,4)\right)}{231 \sqrt{2}} $|$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(-11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)+70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{2}} \left(\sqrt{3} B(5,2)+B(5,4)\right) $|$ \frac{1}{462} \left(66 \sqrt{7} A(1,0)+11 \sqrt{7} A(3,0)-11 \sqrt{210} A(3,2)-25 \sqrt{7} A(5,0)-14 \sqrt{30} A(5,2)+21 \sqrt{10} A(5,4)\right) $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ 0 $|$ \frac{-66 \sqrt{105} A(1,0)-11 \sqrt{105} A(3,0)+5 \left(11 \sqrt{14} A(3,2)+5 \sqrt{105} A(5,0)-70 \sqrt{2} A(5,2)+35 \sqrt{6} A(5,4)\right)}{2310} $|$ \frac{44 \sqrt{7} B(3,2)+35 \left(B(5,2)+\sqrt{3} B(5,4)\right)}{231 \sqrt{2}} $|$ 0 $|$ -\frac{66 \sqrt{35} A(1,0)+11 \sqrt{35} A(3,0)+55 \sqrt{42} A(3,2)-25 \sqrt{35} A(5,0)+70 \sqrt{6} A(5,2)+105 \sqrt{2} A(5,4)}{462 \sqrt{5}} $|$ \frac{1}{11} \sqrt{\frac{5}{2}} \left(B(5,4)-\sqrt{3} B(5,2)\right) $|$ 0 $| |
| | ^$ d_{\text{xy}} $|$ \frac{1}{231} \left(33 \sqrt{7} A(1,0)-22 \sqrt{7} A(3,0)+5 \sqrt{7} A(5,0)-21 \sqrt{10} A(5,4)\right) $|$ 0 $|$ 0 $|$ \frac{2}{3} \sqrt{\frac{2}{7}} B(3,2)-\frac{5}{33} \sqrt{2} B(5,2) $|$ 0 $|$ 0 $|$ -\frac{1}{11} \sqrt{10} B(5,4) $| |
| | |
| | |
| | ### |
| | |
| | </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> | | | | | | |
| | |
| | ### |
