Table of Contents
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Orientation Z
Symmetry Operations
In the C6 Point Group, with orientation Z there are the following symmetry operations
Operator | Orientation |
---|---|
$\text{E}$ | $\{0,0,0\}$ , |
$C_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
$C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
$C_2$ | $\{0,0,1\}$ , |
Different Settings
Character Table
$ $ | $ \text{E} \,{\text{(1)}} $ | $ C_6 \,{\text{(2)}} $ | $ C_3 \,{\text{(2)}} $ | $ C_2 \,{\text{(1)}} $ |
---|---|---|---|---|
$ \text{A} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ |
$ \text{B} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ |
$ E_1 $ | $ 2 $ | $ 1 $ | $ -1 $ | $ -2 $ |
$ E_2 $ | $ 2 $ | $ -1 $ | $ -1 $ | $ 2 $ |
Product Table
$ $ | $ \text{A} $ | $ \text{B} $ | $ E_1 $ | $ E_2 $ |
---|---|---|---|---|
$ \text{A} $ | $ \text{A} $ | $ \text{B} $ | $ E_1 $ | $ E_2 $ |
$ \text{B} $ | $ \text{B} $ | $ \text{A} $ | $ E_2 $ | $ E_1 $ |
$ E_1 $ | $ E_1 $ | $ E_2 $ | $ 2 \text{A}+E_2 $ | $ 2 \text{B}+E_1 $ |
$ E_2 $ | $ E_2 $ | $ E_1 $ | $ 2 \text{B}+E_1 $ | $ 2 \text{A}+E_2 $ |
Sub Groups with compatible settings
Super Groups with compatible settings
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 C6 Point group with orientation Z the form of the expansion coefficients is:
Expansion
$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(1,0) & k=1\land m=0 \\ A(2,0) & k=2\land m=0 \\ A(3,0) & k=3\land m=0 \\ A(4,0) & k=4\land m=0 \\ A(5,0) & k=5\land m=0 \\ A(6,6)-i B(6,6) & k=6\land m=-6 \\ A(6,0) & k=6\land m=0 \\ A(6,6)+i B(6,6) & k=6\land m=6 \end{cases}$$
Input format suitable for Mathematica (Quanty.nb)
- Akm_C6_Z.Quanty.nb
Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[3, 0], k == 3 && m == 0}, {A[4, 0], k == 4 && m == 0}, {A[5, 0], k == 5 && m == 0}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 0], k == 6 && m == 0}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0]
Input format suitable for Quanty
- Akm_C6_Z.Quanty
Akm = {{0, 0, A(0,0)} , {1, 0, A(1,0)} , {2, 0, A(2,0)} , {3, 0, A(3,0)} , {4, 0, A(4,0)} , {5, 0, A(5,0)} , {6, 0, A(6,0)} , {6,-6, A(6,6) + (-I)*(B(6,6))} , {6, 6, A(6,6) + (I)*(B(6,6))} }
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) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ {Y_{-1}^{(1)}} $ | $\color{darkred}{ 0 }$ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ {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 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ {Y_{1}^{(1)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $ 0 $ | $ 0 $ |
$ {Y_{-2}^{(2)}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ {Y_{-1}^{(2)}} $ | $ 0 $ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ {Y_{0}^{(2)}} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ {Y_{1}^{(2)}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ {Y_{2}^{(2)}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ |
$ {Y_{-3}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $ |
$ {Y_{-2}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 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 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 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 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ {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 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ {Y_{1}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 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 $ | $ 0 $ |
$ {Y_{2}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $ | $ 0 $ |
$ {Y_{3}^{(3)}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 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_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 $ |
$ 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 $ |
$ d_{\text{xy}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \frac{i}{\sqrt{2}} $ | $ 0 $ | $ 0 $ | $ 0 $ | $ -\frac{i}{\sqrt{2}} $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ d_{\text{yz}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \frac{i}{\sqrt{2}} $ | $ 0 $ | $ \frac{i}{\sqrt{2}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ d_{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{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_{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}} $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ f_{y\left(3x^2-y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \frac{i}{\sqrt{2}} $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \frac{i}{\sqrt{2}} $ |
$ 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_{y\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 $ | $ \frac{i}{\sqrt{2}} $ | $ 0 $ | $ \frac{i}{\sqrt{2}} $ | $ 0 $ | $ 0 $ |
$ f_{z\left(5z^2-3r^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(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 $ | $ \frac{1}{\sqrt{2}} $ | $ 0 $ | $ -\frac{1}{\sqrt{2}} $ | $ 0 $ | $ 0 $ |
$ f_{z\left(x^2-y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \frac{1}{\sqrt{2}} $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \frac{1}{\sqrt{2}} $ | $ 0 $ |
$ f_{x\left(x^2-3y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \frac{1}{\sqrt{2}} $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ -\frac{1}{\sqrt{2}} $ |
One particle coupling on a basis of symmetry adapted functions
After rotation we find
$ $ | $ \text{s} $ | $ p_y $ | $ p_z $ | $ p_x $ | $ d_{\text{xy}} $ | $ d_{\text{yz}} $ | $ d_{3z^2-r^2} $ | $ d_{\text{xz}} $ | $ d_{x^2-y^2} $ | $ f_{y\left(3x^2-y^2\right)} $ | $ f_{\text{xyz}} $ | $ f_{y\left(5z^2-r^2\right)} $ | $ f_{z\left(5z^2-3r^2\right)} $ | $ f_{x\left(5z^2-r^2\right)} $ | $ f_{z\left(x^2-y^2\right)} $ | $ f_{x\left(x^2-3y^2\right)} $ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$ \text{s} $ | $ \text{Ass}(0,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ p_y $ | $\color{darkred}{ 0 }$ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ p_z $ | $\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$ | $ 0 $ | $ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ p_x $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $ 0 $ | $ 0 $ |
$ d_{\text{xy}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ d_{\text{yz}} $ | $ 0 $ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ d_{3z^2-r^2} $ | $ \frac{\text{Asd}(2,0)}{\sqrt{5}} $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ d_{\text{xz}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ |
$ d_{x^2-y^2} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ |
$ f_{y\left(3x^2-y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $ |
$ f_{\text{xyz}} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ f_{y\left(5z^2-r^2\right)} $ | $\color{darkred}{ 0 }$ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 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 $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ f_{z\left(5z^2-3r^2\right)} $ | $\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $ | $ 0 $ | $ 0 $ | $ 0 $ |
$ f_{x\left(5z^2-r^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 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 $ | $ 0 $ |
$ f_{z\left(x^2-y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $ | $ 0 $ |
$ f_{x\left(x^2-3y^2\right)} $ | $\color{darkred}{ 0 }$ | $ 0 $ | $ 0 $ | $ 0 $ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $\color{darkred}{ 0 }$ | $ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 0 $ | $ 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)+\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $ |
Coupling for a single shell
Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$.
Click on one of the subsections to expand it or
Potential for s orbitals
Potential for p orbitals
Potential for d orbitals
Potential for f orbitals
Coupling between two shells
Click on one of the subsections to expand it or
Potential for s-p orbital mixing
Potential for s-d orbital mixing
Potential for s-f orbital mixing
Potential for p-d orbital mixing
Potential for p-f orbital mixing
Potential for d-f orbital mixing
Table of several point groups
Return to Main page on Point Groups
Nonaxial groups | C1 | Cs | Ci | ||||
---|---|---|---|---|---|---|---|
Cn groups | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
Dn groups | D2 | D3 | D4 | D5 | D6 | D7 | D8 |
Cnv groups | C2v | C3v | C4v | C5v | C6v | C7v | C8v |
Cnh groups | C2h | C3h | C4h | C5h | C6h | ||
Dnh groups | D2h | D3h | D4h | D5h | D6h | D7h | D8h |
Dnd groups | D2d | D3d | D4d | D5d | D6d | D7d | D8d |
Sn groups | S2 | S4 | S6 | S8 | S10 | S12 | |
Cubic groups | T | Th | Td | O | Oh | I | Ih |
Linear groups | C$\infty$v | D$\infty$h |