−Table of Contents

Orientation Zy

Orientation Zy

Symmetry Operations

In the D3h Point Group, with orientation Zy there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{-\sqrt{3},1,0\right\}$ ,
$\sigma _h$	$\{0,0,1\}$ ,
$S_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$\sigma _v$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,

Different Settings

Character Table

	$\text{E} \,{\text{(1)}}$	$C_3 \,{\text{(2)}}$	$C_2 \,{\text{(3)}}$	$\sigma_h \,{\text{(1)}}$	$S_3 \,{\text{(2)}}$	$\sigma_v \,{\text{(3)}}$
$A'_1$	$1$	$1$	$1$	$1$	$1$	$1$
$A'_2$	$1$	$1$	$-1$	$1$	$1$	$-1$
$\text{E'}$	$2$	$-1$	$0$	$2$	$-1$	$0$
$A''_1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
$A''_2$	$1$	$1$	$-1$	$-1$	$-1$	$1$
$\text{E''}$	$2$	$-1$	$0$	$-2$	$1$	$0$

Product Table

	$A'_1$	$A'_2$	$\text{E'}$	$A''_1$	$A''_2$	$\text{E''}$
$A'_1$	$A'_1$	$A'_2$	$\text{E'}$	$A''_1$	$A''_2$	$\text{E''}$
$A'_2$	$A'_2$	$A'_1$	$\text{E'}$	$A''_2$	$A''_1$	$\text{E''}$
$\text{E'}$	$\text{E'}$	$\text{E'}$	$A'_1+A'_2+\text{E'}$	$\text{E''}$	$\text{E''}$	$A''_1+A''_2+\text{E''}$
$A''_1$	$A''_1$	$A''_2$	$\text{E''}$	$A'_1$	$A'_2$	$\text{E'}$
$A''_2$	$A''_2$	$A''_1$	$\text{E''}$	$A'_2$	$A'_1$	$\text{E'}$
$\text{E''}$	$\text{E''}$	$\text{E''}$	$A''_1+A''_2+\text{E''}$	$\text{E'}$	$\text{E'}$	$A'_1+A'_2+\text{E'}$

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 D3h Point group with orientation Zy the form of the expansion coefficients is:

Expansion

$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(2,0) & k=2\land m=0 \\ i B(3,3) & k=3\land (m=-3\lor m=3) \\ A(4,0) & k=4\land m=0 \\ i B(5,3) & k=5\land (m=-3\lor m=3) \\ A(6,6) & k=6\land (m=-6\lor m=6) \\ A(6,0) & k=6\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {I*B[3, 3], k == 3 && (m == -3 || m == 3)}, {A[4, 0], k == 4 && m == 0}, {I*B[5, 3], k == 5 && (m == -3 || m == 3)}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {A[6, 0], k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {3,-3, (I)*(B(3,3))} , 
       {3, 3, (I)*(B(3,3))} , 
       {4, 0, A(4,0)} , 
       {5,-3, (I)*(B(5,3))} , 
       {5, 3, (I)*(B(5,3))} , 
       {6, 0, A(6,0)} , 
       {6,-6, A(6,6)} , 
       {6, 6, A(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.

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$0$	$0$	$\color{darkred}{ -\frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$
${Y_{-1}^{(1)}}$	$\color{darkred}{ 0 }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3}{7} i \text{Bpd}(3,3) }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$0$	$0$
${Y_{0}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$
${Y_{1}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\color{darkred}{ \frac{3}{7} i \text{Bpd}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$
${Y_{-2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{3}{7} i \text{Bpd}(3,3) }$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3)-\frac{5}{33} i \sqrt{2} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$
${Y_{0}^{(2)}}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{33} i \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{2}{33} i \sqrt{5} \text{Bdf}(5,3) }$
${Y_{1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{2}^{(2)}}$	$0$	$\color{darkred}{ -\frac{3}{7} i \text{Bpd}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3)-\frac{5}{33} i \sqrt{2} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-3}^{(3)}}$	$\color{darkred}{ \frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{33} i \sqrt{5} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$	$-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$
${Y_{-2}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$
${Y_{-1}^{(3)}}$	$\color{darkred}{ 0 }$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5}{33} i \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$
${Y_{0}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$
${Y_{1}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ \frac{5}{33} i \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$
${Y_{2}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3)-\frac{4}{33} i \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$
${Y_{3}^{(3)}}$	$\color{darkred}{ \frac{i \text{Bsf}(3,3)}{\sqrt{7}} }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{33} i \sqrt{5} \text{Bdf}(5,3)-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

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

	$\text{s}$	$p_y$	$p_z$	$p_x$	$d_{\text{xy}}$	$d_{\text{yz}}$	$d_{3z^2-r^2}$	$d_{\text{xz}}$	$d_{x^2-y^2}$	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$0$	$0$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bsf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_y$	$\color{darkred}{ 0 }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$0$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$
$p_x$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$
$d_{\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ \frac{2}{33} \sqrt{10} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{x^2-y^2}$	$0$	$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{y\left(3x^2-y^2\right)}$	$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bsf}(3,3) }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{33} \sqrt{10} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{\text{xyz}}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$
$f_{y\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$
$f_{z\left(5z^2-3r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$
$f_{x\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$
$f_{z\left(x^2-y^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$
$f_{x\left(x^2-3y^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)+\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$

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 parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Ea1p} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Ea1p, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{0, 0, Ea1p} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{0}^{(0)}}$
${Y_{0}^{(0)}}$	$\text{Ea1p}$

The Hamiltonian on a basis of symmetric functions

	$\text{s}$
$\text{s}$	$\text{Ea1p}$

Rotation matrix used

	${Y_{0}^{(0)}}$
$\text{s}$	$1$

Irriducible representations and their onsite energy

	$\text{Ea1p}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea2pp}+2 \text{Eep}) & k=0\land m=0 \\ \frac{5 (\text{Ea2pp}-\text{Eep})}{3} & k=2\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea2pp + 2*Eep)/3, k == 0 && m == 0}, {(5*(Ea2pp - Eep))/3, k == 2 && m == 0}}, 0]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{0, 0, (1/3)*(Ea2pp + (2)*(Eep))} , 
       {2, 0, (5/3)*(Ea2pp + (-1)*(Eep))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\text{Eep}$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$\text{Ea2pp}$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$\text{Eep}$

The Hamiltonian on a basis of symmetric functions

	$p_y$	$p_z$	$p_x$
$p_y$	$\text{Eep}$	$0$	$0$
$p_z$	$0$	$\text{Ea2pp}$	$0$
$p_x$	$0$	$0$	$\text{Eep}$

Rotation matrix used

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
$p_y$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$
$p_z$	$0$	$1$	$0$
$p_x$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$

Irriducible representations and their onsite energy

	$\text{Eep}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$
	$\text{Ea2pp}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$
	$\text{Eep}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea1p}+2 (\text{Eep}+\text{Eepp})) & k=0\land m=0 \\ \text{Ea1p}-2 \text{Eep}+\text{Eepp} & k=2\land m=0 \\ 0 & k\neq 4\lor m\neq 0 \\ \frac{3}{5} (3 \text{Ea1p}+\text{Eep}-4 \text{Eepp}) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1p + 2*(Eep + Eepp))/5, k == 0 && m == 0}, {Ea1p - 2*Eep + Eepp, k == 2 && m == 0}, {0, k != 4 || m != 0}}, (3*(3*Ea1p + Eep - 4*Eepp))/5]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{0, 0, (1/5)*(Ea1p + (2)*(Eep + Eepp))} , 
       {2, 0, Ea1p + (-2)*(Eep) + Eepp} , 
       {4, 0, (3/5)*((3)*(Ea1p) + Eep + (-4)*(Eepp))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\text{Eep}$	$0$	$0$	$0$	$0$
${Y_{-1}^{(2)}}$	$0$	$\text{Eepp}$	$0$	$0$	$0$
${Y_{0}^{(2)}}$	$0$	$0$	$\text{Ea1p}$	$0$	$0$
${Y_{1}^{(2)}}$	$0$	$0$	$0$	$\text{Eepp}$	$0$
${Y_{2}^{(2)}}$	$0$	$0$	$0$	$0$	$\text{Eep}$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{xy}}$	$d_{\text{yz}}$	$d_{3z^2-r^2}$	$d_{\text{xz}}$	$d_{x^2-y^2}$
$d_{\text{xy}}$	$\text{Eep}$	$0$	$0$	$0$	$0$
$d_{\text{yz}}$	$0$	$\text{Eepp}$	$0$	$0$	$0$
$d_{3z^2-r^2}$	$0$	$0$	$\text{Ea1p}$	$0$	$0$
$d_{\text{xz}}$	$0$	$0$	$0$	$\text{Eepp}$	$0$
$d_{x^2-y^2}$	$0$	$0$	$0$	$0$	$\text{Eep}$

Rotation matrix used

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
$d_{\text{xy}}$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$
$d_{\text{yz}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$
$d_{3z^2-r^2}$	$0$	$0$	$1$	$0$	$0$
$d_{\text{xz}}$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$0$
$d_{x^2-y^2}$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$\frac{1}{\sqrt{2}}$

Irriducible representations and their onsite energy

	$\text{Eep}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$
	$\text{Eepp}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$
	$\text{Ea1p}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$
	$\text{Eepp}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$
	$\text{Eep}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea1p}+\text{Ea2p}+\text{Ea2pp}+2 \text{Eep}+2 \text{Eepp}) & k=0\land m=0 \\ -\frac{5}{28} (5 \text{Ea1p}+5 \text{Ea2p}-4 \text{Ea2pp}-6 \text{Eep}) & k=2\land m=0 \\ 0 & (k\neq 6\land (k\neq 4\lor m\neq 0))\lor (m\neq -6\land m\neq 0\land m\neq 6) \\ \frac{3}{14} (3 \text{Ea1p}+3 \text{Ea2p}+2 (3 \text{Ea2pp}+\text{Eep}-7 \text{Eepp})) & k=4\land m=0 \\ -\frac{13}{20} \sqrt{\frac{33}{7}} (\text{Ea1p}-\text{Ea2p}) & k=6\land (m=-6\lor m=6) \\ -\frac{13}{140} (\text{Ea1p}+\text{Ea2p}-20 \text{Ea2pp}+30 \text{Eep}-12 \text{Eepp}) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1p + Ea2p + Ea2pp + 2*Eep + 2*Eepp)/7, k == 0 && m == 0}, {(-5*(5*Ea1p + 5*Ea2p - 4*Ea2pp - 6*Eep))/28, k == 2 && m == 0}, {0, (k != 6 && (k != 4 || m != 0)) || (m != -6 && m != 0 && m != 6)}, {(3*(3*Ea1p + 3*Ea2p + 2*(3*Ea2pp + Eep - 7*Eepp)))/14, k == 4 && m == 0}, {(-13*Sqrt[33/7]*(Ea1p - Ea2p))/20, k == 6 && (m == -6 || m == 6)}}, (-13*(Ea1p + Ea2p - 20*Ea2pp + 30*Eep - 12*Eepp))/140]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{0, 0, (1/7)*(Ea1p + Ea2p + Ea2pp + (2)*(Eep) + (2)*(Eepp))} , 
       {2, 0, (-5/28)*((5)*(Ea1p) + (5)*(Ea2p) + (-4)*(Ea2pp) + (-6)*(Eep))} , 
       {4, 0, (3/14)*((3)*(Ea1p) + (3)*(Ea2p) + (2)*((3)*(Ea2pp) + Eep + (-7)*(Eepp)))} , 
       {6, 0, (-13/140)*(Ea1p + Ea2p + (-20)*(Ea2pp) + (30)*(Eep) + (-12)*(Eepp))} , 
       {6,-6, (-13/20)*((sqrt(33/7))*(Ea1p + (-1)*(Ea2p)))} , 
       {6, 6, (-13/20)*((sqrt(33/7))*(Ea1p + (-1)*(Ea2p)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\frac{\text{Ea1p}+\text{Ea2p}}{2}$	$0$	$0$	$0$	$0$	$0$	$\frac{\text{Ea1p}-\text{Ea2p}}{2}$
${Y_{-2}^{(3)}}$	$0$	$\text{Eepp}$	$0$	$0$	$0$	$0$	$0$
${Y_{-1}^{(3)}}$	$0$	$0$	$\text{Eep}$	$0$	$0$	$0$	$0$
${Y_{0}^{(3)}}$	$0$	$0$	$0$	$\text{Ea2pp}$	$0$	$0$	$0$
${Y_{1}^{(3)}}$	$0$	$0$	$0$	$0$	$\text{Eep}$	$0$	$0$
${Y_{2}^{(3)}}$	$0$	$0$	$0$	$0$	$0$	$\text{Eepp}$	$0$
${Y_{3}^{(3)}}$	$\frac{\text{Ea1p}-\text{Ea2p}}{2}$	$0$	$0$	$0$	$0$	$0$	$\frac{\text{Ea1p}+\text{Ea2p}}{2}$

The Hamiltonian on a basis of symmetric functions

	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$f_{y\left(3x^2-y^2\right)}$	$\text{Ea1p}$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{\text{xyz}}$	$0$	$\text{Eepp}$	$0$	$0$	$0$	$0$	$0$
$f_{y\left(5z^2-r^2\right)}$	$0$	$0$	$\text{Eep}$	$0$	$0$	$0$	$0$
$f_{z\left(5z^2-3r^2\right)}$	$0$	$0$	$0$	$\text{Ea2pp}$	$0$	$0$	$0$
$f_{x\left(5z^2-r^2\right)}$	$0$	$0$	$0$	$0$	$\text{Eep}$	$0$	$0$
$f_{z\left(x^2-y^2\right)}$	$0$	$0$	$0$	$0$	$0$	$\text{Eepp}$	$0$
$f_{x\left(x^2-3y^2\right)}$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Ea2p}$

Rotation matrix used

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$f_{y\left(3x^2-y^2\right)}$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$0$	$0$	$\frac{i}{\sqrt{2}}$
$f_{\text{xyz}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$0$	$0$	$-\frac{i}{\sqrt{2}}$	$0$
$f_{y\left(5z^2-r^2\right)}$	$0$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$0$	$0$
$f_{z\left(5z^2-3r^2\right)}$	$0$	$0$	$0$	$1$	$0$	$0$	$0$
$f_{x\left(5z^2-r^2\right)}$	$0$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$0$	$0$
$f_{z\left(x^2-y^2\right)}$	$0$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$\frac{1}{\sqrt{2}}$	$0$
$f_{x\left(x^2-3y^2\right)}$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$0$	$0$	$-\frac{1}{\sqrt{2}}$

Irriducible representations and their onsite energy

	$\text{Ea1p}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$
	$\text{Eepp}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$
	$\text{Eep}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$
	$\text{Ea2pp}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$
	$\text{Eep}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$
	$\text{Eepp}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$
	$\text{Ea2p}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 2\lor m\neq 0 \\ A(2,0) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, A[2, 0]]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{2, 0, A(2,0)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{0}^{(0)}}$	$0$	$0$	$\frac{A(2,0)}{\sqrt{5}}$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{xy}}$	$d_{\text{yz}}$	$d_{3z^2-r^2}$	$d_{\text{xz}}$	$d_{x^2-y^2}$
$\text{s}$	$0$	$0$	$\frac{A(2,0)}{\sqrt{5}}$	$0$	$0$

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -3\land m\neq 3) \\ i B(3,3) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != 3)}}, I*B[3, 3]]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{3,-3, (I)*(B(3,3))} , 
       {3, 3, (I)*(B(3,3))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$-\frac{i B(3,3)}{\sqrt{7}}$	$0$	$0$	$0$	$0$	$0$	$-\frac{i B(3,3)}{\sqrt{7}}$

The Hamiltonian on a basis of symmetric functions

	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$\text{s}$	$\sqrt{\frac{2}{7}} B(3,3)$	$0$	$0$	$0$	$0$	$0$	$0$

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -3\land m\neq 3) \\ i B(3,3) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != 3)}}, I*B[3, 3]]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{3,-3, (I)*(B(3,3))} , 
       {3, 3, (I)*(B(3,3))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$0$	$0$	$\frac{3}{7} i B(3,3)$
${Y_{0}^{(1)}}$	$0$	$0$	$0$	$0$	$0$
${Y_{1}^{(1)}}$	$\frac{3}{7} i B(3,3)$	$0$	$0$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$d_{\text{xy}}$	$d_{\text{yz}}$	$d_{3z^2-r^2}$	$d_{\text{xz}}$	$d_{x^2-y^2}$
$p_y$	$0$	$0$	$0$	$0$	$\frac{3}{7} B(3,3)$
$p_z$	$0$	$0$	$0$	$0$	$0$
$p_x$	$\frac{3}{7} B(3,3)$	$0$	$0$	$0$	$0$

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & (k\neq 2\land k\neq 4)\lor m\neq 0 \\ A(2,0) & k=2\land m=0 \\ A(4,0) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || m != 0}, {A[2, 0], k == 2 && m == 0}}, A[4, 0]]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{2, 0, A(2,0)} , 
       {4, 0, A(4,0)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$	$0$	$0$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$0$	$0$	$\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$	$0$	$0$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$0$	$0$	$\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$p_y$	$0$	$0$	$\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$	$0$	$0$	$0$	$0$
$p_z$	$0$	$0$	$0$	$\frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}}$	$0$	$0$	$0$
$p_x$	$0$	$0$	$0$	$0$	$\frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0))$	$0$	$0$

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & (k\neq 3\land k\neq 5)\lor (m\neq -3\land m\neq 3) \\ i B(3,3) & k=3\land (m=-3\lor m=3) \\ i B(5,3) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3h_Zy.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 3 && k != 5) || (m != -3 && m != 3)}, {I*B[3, 3], k == 3 && (m == -3 || m == 3)}}, I*B[5, 3]]

Input format suitable for Quanty

Akm_D3h_Zy.Quanty

Akm = {{3,-3, (I)*(B(3,3))} , 
       {3, 3, (I)*(B(3,3))} , 
       {5,-3, (I)*(B(5,3))} , 
       {5, 3, (I)*(B(5,3))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-2}^{(2)}}$	$0$	$0$	$0$	$0$	$\frac{1}{231} i \sqrt{2} \left(11 \sqrt{7} B(3,3)-35 B(5,3)\right)$	$0$	$0$
${Y_{-1}^{(2)}}$	$0$	$0$	$0$	$0$	$0$	$\frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)+28 B(5,3)\right)$	$0$
${Y_{0}^{(2)}}$	$\frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)-14 B(5,3)\right)$	$0$	$0$	$0$	$0$	$0$	$\frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)-14 B(5,3)\right)$
${Y_{1}^{(2)}}$	$0$	$\frac{1}{231} i \sqrt{5} \left(11 \sqrt{7} B(3,3)+28 B(5,3)\right)$	$0$	$0$	$0$	$0$	$0$
${Y_{2}^{(2)}}$	$0$	$0$	$\frac{1}{231} i \sqrt{2} \left(11 \sqrt{7} B(3,3)-35 B(5,3)\right)$	$0$	$0$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{y\left(3x^2-y^2\right)}$	$f_{\text{xyz}}$	$f_{y\left(5z^2-r^2\right)}$	$f_{z\left(5z^2-3r^2\right)}$	$f_{x\left(5z^2-r^2\right)}$	$f_{z\left(x^2-y^2\right)}$	$f_{x\left(x^2-3y^2\right)}$
$d_{\text{xy}}$	$0$	$0$	$0$	$0$	$\frac{5}{33} \sqrt{2} B(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} B(3,3)$	$0$	$0$
$d_{\text{yz}}$	$0$	$0$	$0$	$0$	$0$	$\frac{1}{3} \sqrt{\frac{5}{7}} B(3,3)+\frac{4}{33} \sqrt{5} B(5,3)$	$0$
$d_{3z^2-r^2}$	$\frac{2}{33} \sqrt{10} B(5,3)-\frac{1}{3} \sqrt{\frac{10}{7}} B(3,3)$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{\text{xz}}$	$0$	$\frac{1}{3} \sqrt{\frac{5}{7}} B(3,3)+\frac{4}{33} \sqrt{5} B(5,3)$	$0$	$0$	$0$	$0$	$0$
$d_{x^2-y^2}$	$0$	$0$	$\frac{5}{33} \sqrt{2} B(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} B(3,3)$	$0$	$0$	$0$	$0$

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups	C₁	C_s	C_i
C_n groups	C₂	C₃	C₄	C₅	C₆	C₇	C₈
D_n groups	D₂	D₃	D₄	D₅	D₆	D₇	D₈
C_nv groups	C_2v	C_3v	C_4v	C_5v	C_6v	C_7v	C_8v
C_nh groups	C_2h	C_3h	C_4h	C_5h	C_6h
D_nh groups	D_2h	D_3h	D_4h	D_5h	D_6h	D_7h	D_8h
D_nd groups	D_2d	D_3d	D_4d	D_5d	D_6d	D_7d	D_8d
S_n groups	S₂	S₄	S₆	S₈	S₁₀	S₁₂
Cubic groups	T	T_h	T_d	O	O_h	I	I_h
Linear groups	C_{$\infty$ v}	D_{$\infty$ h}