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 1

• 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 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 Cs Point group with orientation Z the form of the expansion coefficients is:

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

Akm_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))} }

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 we can choose a basis of spherical harmonics times some radial function $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice we can separate the radial part from the angular part for the evaluation of the operator. With the definition $$A_{l'',l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle$$ we can express the operator as $$O = \sum_{n'',l'',m'',n',l',m',k,m} A_{l'',l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ The coefficient in front of the creation and annihilation operators $$\sum_{k,m} A_{l'',l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle$$ is shown in the table below. 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.

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