Orientation Z

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

• Point Group Cs with orientation X
• Point Group Cs with orientation Y
• Point Group Cs with orientation Z

• Point Group C1 with orientation

• Point Group C2h with orientation Z
• Point Group C3h with orientation Z
• Point Group C4h with orientation Z
• Point Group C6h with orientation Z
• Point Group D2h with orientation XYZ
• Point Group D3h with orientation Zx
• Point Group D3h with orientation Zy
• Point Group D4h with orientation Zxy
• Point Group D6h with orientation Zx
• Point Group D6h with orientation Zy
• Point Group Oh with orientation XYZ
• Point Group Th with orientation xyz

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:

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

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))} }

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