+Table of Contents

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Orientation Z

Symmetry Operations

In the Cs Point Group, with orientation Z there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$\sigma _h$	$\{0,0,1\}$ ,

Different Settings

Character Table

	$\text{E} \,{\text{(1)}}$	$\sigma_h \,{\text{(1)}}$
$\text{A'}$	$1$	$1$
$\text{A''}$	$1$	$-1$

Product Table

	$\text{A'}$	$\text{A''}$
$\text{A'}$	$\text{A'}$	$\text{A''}$
$\text{A''}$	$\text{A''}$	$\text{A'}$

Sub Groups with compatible settings

Point Group C1 with orientation

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Any potential (function) can be written in spherical coordinates as a sum over spherical harmonics $V(\vec{r}) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$ . With $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics $C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$ . The allowed expansion coefficients $A_{k,m}(r)$ , or once evaluated for a given radial wave-function $A_{k,m}=\langle\psi(r)|A_{k,m}(r)|\psi(r)\rangle$ , such that $V(\vec{r}) is invariant under all symmetry operations of the Cs Point group with orientation Z are:

Input format suitable for Mathematica (Quanty.nb)

$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ A(2,0) & k=2\land m=0 \\ A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ A(4,0) & k=4\land m=0 \\ A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ A(6,0) & k=6\land m=0 \\ A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 \end{cases}$

Input format suitable for Quanty

Akm_Cs_Z.Quanty

Akm = {{0, 0, A(0,0)} , 
       {1,-1, (-1)*(A(1,1)) + ((+1*I))*(Ap(1,1))} , 
       {1, 1, A(1,1) + ((+1*I))*(Ap(1,1))} , 
       {2, 0, A(2,0)} , 
       {2,-2, A(2,2) + ((+-1*I))*(Ap(2,2))} , 
       {2, 2, A(2,2) + ((+1*I))*(Ap(2,2))} , 
       {3,-1, (-1)*(A(3,1)) + ((+1*I))*(Ap(3,1))} , 
       {3, 1, A(3,1) + ((+1*I))*(Ap(3,1))} , 
       {3,-3, (-1)*(A(3,3)) + ((+1*I))*(Ap(3,3))} , 
       {3, 3, A(3,3) + ((+1*I))*(Ap(3,3))} , 
       {4, 0, A(4,0)} , 
       {4,-2, A(4,2) + ((+-1*I))*(Ap(4,2))} , 
       {4, 2, A(4,2) + ((+1*I))*(Ap(4,2))} , 
       {4,-4, A(4,4) + ((+-1*I))*(Ap(4,4))} , 
       {4, 4, A(4,4) + ((+1*I))*(Ap(4,4))} , 
       {5,-1, (-1)*(A(5,1)) + ((+1*I))*(Ap(5,1))} , 
       {5, 1, A(5,1) + ((+1*I))*(Ap(5,1))} , 
       {5,-3, (-1)*(A(5,3)) + ((+1*I))*(Ap(5,3))} , 
       {5, 3, A(5,3) + ((+1*I))*(Ap(5,3))} , 
       {5,-5, (-1)*(A(5,5)) + ((+1*I))*(Ap(5,5))} , 
       {5, 5, A(5,5) + ((+1*I))*(Ap(5,5))} , 
       {6, 0, A(6,0)} , 
       {6,-2, A(6,2) + ((+-1*I))*(Ap(6,2))} , 
       {6, 2, A(6,2) + ((+1*I))*(Ap(6,2))} , 
       {6,-4, A(6,4) + ((+-1*I))*(Ap(6,4))} , 
       {6, 4, A(6,4) + ((+1*I))*(Ap(6,4))} , 
       {6,-6, A(6,6) + ((+-1*I))*(Ap(6,6))} , 
       {6, 6, A(6,6) + ((+1*I))*(Ap(6,6))} }

One particle coupling on a basis of spherical harmonics

	$\color{darkred}{\text{Subsuperscript}[\text{Y},0,\text{(0)}]}$	$\text{Subsuperscript}[\text{Y},-1,\text{(1)}]$	$\text{Subsuperscript}[\text{Y},0,\text{(1)}]$	$\text{Subsuperscript}[\text{Y},1,\text{(1)}]$	$\text{Subsuperscript}[\text{Y},-2,\text{(2)}]$	$\text{Subsuperscript}[\text{Y},-1,\text{(2)}]$	$\text{Subsuperscript}[\text{Y},0,\text{(2)}]$	$\text{Subsuperscript}[\text{Y},1,\text{(2)}]$	$\text{Subsuperscript}[\text{Y},2,\text{(2)}]$	$\text{Subsuperscript}[\text{Y},-3,\text{(3)}]$	$\text{Subsuperscript}[\text{Y},-2,\text{(3)}]$	$\text{Subsuperscript}[\text{Y},-1,\text{(3)}]$	$\text{Subsuperscript}[\text{Y},0,\text{(3)}]$	$\text{Subsuperscript}[\text{Y},1,\text{(3)}]$	$\text{Subsuperscript}[\text{Y},2,\text{(3)}]$	$\text{Subsuperscript}[\text{Y},3,\text{(3)}]$
$\text{Subsuperscript}[\text{Y},0,\text{(0)}]$	$\text{Ass}(0,0)$	$-\frac{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}}$	$0$	$-\frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}}$	$\frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}}$	$0$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$0$	$\frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}}$	$-\frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}}$	$0$	$-\frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}}$	$0$	$-\frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}}$	$0$	$-\frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}}$
$\text{Subsuperscript}[\text{Y},-1,\text{(1)}]$	$\frac{-\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}}$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$-\frac{1}{5} \sqrt{6} (\text{App}(2,2)-i \text{Bpp}(2,2))$	$\frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1))$	$0$	$\frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}$	$0$	$\frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3))$	$\frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}}$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$\frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2))$	$0$	$-\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}}$
$\text{Subsuperscript}[\text{Y},0,\text{(1)}]$	$0$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$0$	$0$	$-\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$0$	$-\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$0$	$0$	$\sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}}$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}}$	$0$
$\text{Subsuperscript}[\text{Y},1,\text{(1)}]$	$\frac{\text{Asp}(1,1)+i \text{Bsp}(1,1)}{\sqrt{3}}$	$-\frac{1}{5} \sqrt{6} (\text{App}(2,2)+i \text{Bpp}(2,2))$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3))$	$0$	$\frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))-\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}$	$0$	$\frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))-\sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1))$	$-\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}}$	$0$	$\frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2))$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$\frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}}$
$\text{Subsuperscript}[\text{Y},-2,\text{(2)}]$	$\frac{\text{Asd}(2,2)-i \text{Bsd}(2,2)}{\sqrt{5}}$	$\sqrt{\frac{2}{5}} (-\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$0$	$-\frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3))$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$0$	$\frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2))$	$0$	$\frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)-i \text{Bdd}(4,4))$	$-\sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$-\frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}}$	$0$	$\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$	$-\frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5))$
$\text{Subsuperscript}[\text{Y},-1,\text{(2)}]$	$0$	$0$	$\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$-\frac{1}{7} \sqrt{6} (\text{Add}(2,2)-i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)-i \text{Bdd}(4,2))$	$0$	$0$	$-\sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$-\sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}}$	$0$	$\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$
$\text{Subsuperscript}[\text{Y},0,\text{(2)}]$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$0$	$\frac{-\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{15}}-\frac{3}{7} \sqrt{\frac{2}{5}} (-\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$\frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2))$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$\frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)-i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)-i \text{Bdd}(2,2))$	$\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$	$-\sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$-\sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))$
$\text{Subsuperscript}[\text{Y},1,\text{(2)}]$	$0$	$0$	$\frac{\text{Apd}(1,1)+i \text{Bpd}(1,1)}{\sqrt{5}}+\frac{2}{7} \sqrt{\frac{6}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$0$	$0$	$-\frac{1}{7} \sqrt{6} (\text{Add}(2,2)+i \text{Bdd}(2,2))-\frac{2}{21} \sqrt{10} (\text{Add}(4,2)+i \text{Bdd}(4,2))$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$	$-\sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}}$	$0$	$-\sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$
$\text{Subsuperscript}[\text{Y},2,\text{(2)}]$	$\frac{\text{Asd}(2,2)+i \text{Bsd}(2,2)}{\sqrt{5}}$	$-\frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3))$	$0$	$\sqrt{\frac{2}{5}} (\text{Apd}(1,1)+i \text{Bpd}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (\text{Apd}(3,1)+i \text{Bpd}(3,1))$	$\frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(4,4))$	$0$	$\frac{1}{7} \sqrt{\frac{5}{3}} (\text{Add}(4,2)+i \text{Bdd}(4,2))-\frac{2}{7} (\text{Add}(2,2)+i \text{Bdd}(2,2))$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$-\frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5))$	$0$	$\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$	$-\frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}}$	$0$	$-\sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1))$
$\text{Subsuperscript}[\text{Y},-3,\text{(3)}]$	$\frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}}$	$\frac{3 (\text{Apf}(2,2)-i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)-i \text{Bpf}(4,2)}{3 \sqrt{21}}$	$0$	$-\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}}$	$\sqrt{\frac{3}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$\frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))$	$0$	$\frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5))$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2))$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4))$	$0$	$-\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6))$
$\text{Subsuperscript}[\text{Y},-2,\text{(3)}]$	$0$	$0$	$\sqrt{\frac{3}{35}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)-i \text{Bpf}(4,2))}{3 \sqrt{7}}$	$0$	$0$	$\sqrt{\frac{2}{7}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$	$-\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2))$	$0$	$\frac{1}{33} \sqrt{70} (\text{Aff}(4,4)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(6,4))$	$0$
$\text{Subsuperscript}[\text{Y},-1,\text{(3)}]$	$\frac{-\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}}$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$\frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)-i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)-i \text{Bpf}(4,2))$	$\frac{\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}}$	$0$	$\sqrt{\frac{6}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{-\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (-\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$\frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))$	$-\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2))$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$-\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)-i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)-i \text{Bff}(6,2))$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4))$
$\text{Subsuperscript}[\text{Y},0,\text{(3)}]$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$\sqrt{\frac{3}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}}$	$0$	$\sqrt{\frac{3}{35}} (-\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{35}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))-\frac{20 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{33 \sqrt{7}}$	$0$	$0$	$-\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2))$	$0$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$-\frac{2 (\text{Aff}(2,2)-i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)-i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)-i \text{Bff}(6,2))$	$0$
$\text{Subsuperscript}[\text{Y},1,\text{(3)}]$	$\frac{\text{Asf}(3,1)+i \text{Bsf}(3,1)}{\sqrt{7}}$	$\frac{1}{5} \sqrt{\frac{3}{7}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Apf}(4,2)+i \text{Bpf}(4,2))$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))$	$0$	$\sqrt{\frac{6}{35}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{35}}+\frac{5}{11} \sqrt{\frac{2}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$\frac{-\text{Adf}(1,1)+i \text{Bdf}(1,1)}{\sqrt{35}}-2 \sqrt{\frac{2}{105}} (-\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{5 (-\text{Adf}(5,1)+i \text{Bdf}(5,1))}{11 \sqrt{21}}$	$\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4))$	$0$	$-\frac{2}{5} \sqrt{\frac{2}{3}} (\text{Aff}(2,2)+i \text{Bff}(2,2))-\frac{2}{33} \sqrt{10} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{143} \sqrt{\frac{35}{3}} (\text{Aff}(6,2)+i \text{Bff}(6,2))$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)-i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)-i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)-i \text{Bff}(6,2))$
$\text{Subsuperscript}[\text{Y},2,\text{(3)}]$	$0$	$0$	$\sqrt{\frac{3}{35}} (\text{Apf}(2,2)+i \text{Bpf}(2,2))+\frac{2 (\text{Apf}(4,2)+i \text{Bpf}(4,2))}{3 \sqrt{7}}$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))$	$0$	$\sqrt{\frac{2}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))+\frac{\text{Adf}(3,1)+i \text{Bdf}(3,1)}{\sqrt{21}}-\frac{2}{11} \sqrt{\frac{10}{21}} (\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$0$	$0$	$\frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(6,4))$	$0$	$-\frac{2 (\text{Aff}(2,2)+i \text{Bff}(2,2))}{3 \sqrt{5}}-\frac{\text{Aff}(4,2)+i \text{Bff}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (\text{Aff}(6,2)+i \text{Bff}(6,2))$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$
$\text{Subsuperscript}[\text{Y},3,\text{(3)}]$	$\frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}}$	$-\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(4,4))}{3 \sqrt{3}}$	$0$	$\frac{3 (\text{Apf}(2,2)+i \text{Bpf}(2,2))}{\sqrt{35}}-\frac{\text{Apf}(4,2)+i \text{Bpf}(4,2)}{3 \sqrt{21}}$	$\frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5))$	$0$	$\frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))$	$0$	$\sqrt{\frac{3}{7}} (\text{Adf}(1,1)+i \text{Bdf}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,1)+i \text{Bdf}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (\text{Adf}(5,1)+i \text{Bdf}(5,1))$	$-\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6))$	$0$	$\frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(6,4))$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{5}} (\text{Aff}(2,2)+i \text{Bff}(2,2))+\frac{1}{11} \sqrt{6} (\text{Aff}(4,2)+i \text{Bff}(4,2))-\frac{10}{429} \sqrt{7} (\text{Aff}(6,2)+i \text{Bff}(6,2))$	$0$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

Rotation matrix to symmetry adapted functions (choice is not unique)

One particle coupling on a basis of symmetry adapted functions

Potential for s orbitals

Potential for p orbitals

Potential for d orbitals

Potential for f orbitals

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	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}

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