~~CLOSETOC~~

====== Orientation Zy ======

===== Symmetry Operations =====

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In the D5h Point Group, with orientation Zy there are the following symmetry operations

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{{:physics_chemistry:pointgroup:d5h_zy.png}}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_5$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_5^2$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{0,1,0\}$ , $\left\{-\sqrt{5+2 \sqrt{5}},1,0\right\}$ , $\left\{1,\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{1,-\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{\sqrt{5+2 \sqrt{5}},1,0\right\}$ , |
^ $\sigma _h$ | $\{0,0,1\}$ , |
^ $S_5$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $S_5^3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _v$ | $\{1,0,0\}$ , $\left\{1,\sqrt{5+2 \sqrt{5}},0\right\}$ , $\left\{-\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{1,-\sqrt{5+2 \sqrt{5}},0\right\}$ , |

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===== Different Settings =====

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  * [[physics_chemistry:point_groups:d5h:orientation_zx|Point Group D5h with orientation Zx]]
  * [[physics_chemistry:point_groups:d5h:orientation_zy|Point Group D5h with orientation Zy]]

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===== Character Table =====

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|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_5 \,{\text{(2)}} $  ^  $ C_5^2{} \,{\text{(2)}} $  ^  $ C_2 \,{\text{(5)}} $  ^  $ \sigma_h \,{\text{(1)}} $  ^  $ S_5 \,{\text{(2)}} $  ^  $ S_5^3{} \,{\text{(2)}} $  ^  $ \sigma_v \,{\text{(5)}} $  ^
^ $ A'_1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
^ $ A''_1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |
^ $ A'_2 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |
^ $ A''_2 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |
^ $ E'_1 $ |  $ 2 $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ 0 $ |  $ 2 $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ 0 $ |
^ $ E''_1 $ |  $ 2 $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ 0 $ |  $ -2 $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ 0 $ |
^ $ E'_2 $ |  $ 2 $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ 0 $ |  $ 2 $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ 0 $ |
^ $ E''_2 $ |  $ 2 $ |  $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ |  $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ |  $ 0 $ |  $ -2 $ |  $ \frac{1}{2} \left(1+\sqrt{5}\right) $ |  $ \frac{1}{2} \left(1-\sqrt{5}\right) $ |  $ 0 $ |

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===== Product Table =====

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|  $  $  ^  $ A'_1 $  ^  $ A''_1 $  ^  $ A'_2 $  ^  $ A''_2 $  ^  $ E'_1 $  ^  $ E''_1 $  ^  $ E'_2 $  ^  $ E''_2 $  ^
^ $ A'_1 $  | $ A'_1 $  | $ A''_1 $  | $ A'_2 $  | $ A''_2 $  | $ E'_1 $  | $ E''_1 $  | $ E'_2 $  | $ E''_2 $  |
^ $ A''_1 $  | $ A''_1 $  | $ A'_1 $  | $ A''_2 $  | $ A'_2 $  | $ E''_1 $  | $ E'_1 $  | $ E''_2 $  | $ E'_2 $  |
^ $ A'_2 $  | $ A'_2 $  | $ A''_2 $  | $ A'_1 $  | $ A''_1 $  | $ E'_1 $  | $ E''_1 $  | $ E'_2 $  | $ E''_2 $  |
^ $ A''_2 $  | $ A''_2 $  | $ A'_2 $  | $ A''_1 $  | $ A'_1 $  | $ E''_1 $  | $ E'_1 $  | $ E''_2 $  | $ E'_2 $  |
^ $ E'_1 $  | $ E'_1 $  | $ E''_1 $  | $ E'_1 $  | $ E''_1 $  | $ A'_1+A'_2+E'_2 $  | $ A''_1+A''_2+E''_2 $  | $ E'_1+E'_2 $  | $ E''_1+E''_2 $  |
^ $ E''_1 $  | $ E''_1 $  | $ E'_1 $  | $ E''_1 $  | $ E'_1 $  | $ A''_1+A''_2+E''_2 $  | $ A'_1+A'_2+E'_2 $  | $ E''_1+E''_2 $  | $ E'_1+E'_2 $  |
^ $ E'_2 $  | $ E'_2 $  | $ E''_2 $  | $ E'_2 $  | $ E''_2 $  | $ E'_1+E'_2 $  | $ E''_1+E''_2 $  | $ A'_1+A'_2+E'_1 $  | $ A''_1+A''_2+E''_1 $  |
^ $ E''_2 $  | $ E''_2 $  | $ E'_2 $  | $ E''_2 $  | $ E'_2 $  | $ E''_1+E''_2 $  | $ E'_1+E'_2 $  | $ A''_1+A''_2+E''_1 $  | $ A'_1+A'_2+E'_1 $  |

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===== Sub Groups with compatible settings =====

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  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
  * [[physics_chemistry:point_groups:c2:orientation_y|Point Group C2 with orientation Y]]
  * [[physics_chemistry:point_groups:c5:orientation_z|Point Group C5 with orientation Z]]
  * [[physics_chemistry:point_groups:cs:orientation_x|Point Group Cs with orientation X]]
  * [[physics_chemistry:point_groups:cs:orientation_z|Point Group Cs with orientation Z]]

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===== Super Groups with compatible settings =====

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===== Invariant Potential expanded on renormalized spherical Harmonics =====

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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 D5h Point group with orientation Zy the form of the expansion coefficients is:

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==== Expansion ====

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

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==== Input format suitable for Mathematica (Quanty.nb) ====

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<code Quanty Akm_D5h_Zy.Quanty.nb>

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

</code>

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==== Input format suitable for Quanty ====

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<code Quanty Akm_D5h_Zy.Quanty>

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

</code>

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==== One particle coupling on a basis of spherical harmonics ====

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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 $A_{n''l'',n'l'}(k,m)$


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we can express the operator as 
$$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$


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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.

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|  $  $  ^  $ {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}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\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}{ 0 }$|$\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}{ 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}{ 0 }$|$\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}{ 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}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|
^$ {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}{ 0 }$|$\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}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|
^$ {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}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 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}{ -\frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\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}{ \frac{5}{11} i \sqrt{\frac{2}{3}} \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 $|$ 0 $|$ 0 $|$ 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}{ 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}{ 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}{ 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}{ 0 }$|$\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}{ 0 }$|$\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}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{5}{11} i \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\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) $|


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

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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

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|  $  $  ^  $ {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}{ 0 }$|$\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}{ 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}{ 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}{ 0 }$|$\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}{ 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}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|
^$ 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}{ 0 }$|$\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}{ 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 $|$ 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}{ 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}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\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}{ \frac{5}{11} \sqrt{\frac{2}{3}} \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 $|$ 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}{ 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}{ 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}{ 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}{ 0 }$|$\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}{ 0 }$|$\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}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\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) $|


###

===== 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 <hiddenSwitch expand all> 

###

==== Potential for s orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

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

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

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

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

Akm = {{0, 0, Ea1p} }

</code>

###

</hidden>
<hidden **The Hamiltonian on a basis of spherical Harmonics** >

###

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


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ \text{s} $  ^
^$ \text{s} $|$ \text{Ea1p} $|


###

</hidden>
<hidden **Rotation matrix used** >

###

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


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Ea1p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_0_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |


###

</hidden>
==== Potential for p orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{3} (\text{Ea2pp}+2 \text{Ee1p}) & k=0\land m=0 \\
 \frac{5 (\text{Ea2pp}-\text{Ee1p})}{3} & k=2\land m=0
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Ea2pp + 2*Ee1p)/3, k == 0 && m == 0}, {(5*(Ea2pp - Ee1p))/3, k == 2 && m == 0}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

Akm = {{0, 0, (1/3)*(Ea2pp + (2)*(Ee1p))} , 
       {2, 0, (5/3)*(Ea2pp + (-1)*(Ee1p))} }

</code>

###

</hidden>
<hidden **The Hamiltonian on a basis of spherical Harmonics** >

###

|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
^$ {Y_{-1}^{(1)}} $|$ \text{Ee1p} $|$ 0 $|$ 0 $|
^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea2pp} $|$ 0 $|
^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Ee1p} $|


###

</hidden>
<hidden **The Hamiltonian on a basis of symmetric functions** >

###

|  $  $  ^  $ p_y $  ^  $ p_z $  ^  $ p_x $  ^
^$ p_y $|$ \text{Ee1p} $|$ 0 $|$ 0 $|
^$ p_z $|$ 0 $|$ \text{Ea2pp} $|$ 0 $|
^$ p_x $|$ 0 $|$ 0 $|$ \text{Ee1p} $|


###

</hidden>
<hidden **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}} $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Ee1p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Ea2pp}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
^ ^$$\text{Ee1p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_1_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |


###

</hidden>
==== Potential for d orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{5} (\text{Ea1p}+2 (\text{Ee1pp}+\text{Ee2p})) & k=0\land m=0 \\
 \text{Ea1p}+\text{Ee1pp}-2 \text{Ee2p} & k=2\land m=0 \\
 \frac{3}{5} (3 \text{Ea1p}-4 \text{Ee1pp}+\text{Ee2p}) & k=4\land m=0
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Ea1p + 2*(Ee1pp + Ee2p))/5, k == 0 && m == 0}, {Ea1p + Ee1pp - 2*Ee2p, k == 2 && m == 0}, {(3*(3*Ea1p - 4*Ee1pp + Ee2p))/5, k == 4 && m == 0}}, 0]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

Akm = {{0, 0, (1/5)*(Ea1p + (2)*(Ee1pp + Ee2p))} , 
       {2, 0, Ea1p + Ee1pp + (-2)*(Ee2p)} , 
       {4, 0, (3/5)*((3)*(Ea1p) + (-4)*(Ee1pp) + Ee2p)} }

</code>

###

</hidden>
<hidden **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{Ee2p} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Ee1pp} $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea1p} $|$ 0 $|$ 0 $|
^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1pp} $|$ 0 $|
^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2p} $|


###

</hidden>
<hidden **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{Ee2p} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{\text{yz}} $|$ 0 $|$ \text{Ee1pp} $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Ea1p} $|$ 0 $|$ 0 $|
^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1pp} $|$ 0 $|
^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2p} $|


###

</hidden>
<hidden **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}} $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Ee2p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |
^ ^$$\text{Ee1pp}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
^ ^$$\text{Ea1p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
^ ^$$\text{Ee1pp}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
^ ^$$\text{Ee2p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |


###

</hidden>
==== Potential for f orbitals ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

 $$A_{k,m} = \begin{cases}
 \frac{1}{7} (\text{Ea2pp}+2 (\text{Ee1p}+\text{Ee2p}+\text{Ee2pp})) & k=0\land m=0 \\
 \frac{5}{14} (2 \text{Ea2pp}+3 \text{Ee1p}-5 \text{Ee2p}) & k=2\land m=0 \\
 \frac{3}{7} (3 \text{Ea2pp}+\text{Ee1p}+3 \text{Ee2p}-7 \text{Ee2pp}) & k=4\land m=0 \\
 0 & k\neq 6\lor m\neq 0 \\
 \frac{13}{70} (10 \text{Ea2pp}-15 \text{Ee1p}-\text{Ee2p}+6 \text{Ee2pp}) & \text{True}
\end{cases}$$

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

Akm[k_,m_]:=Piecewise[{{(Ea2pp + 2*(Ee1p + Ee2p + Ee2pp))/7, k == 0 && m == 0}, {(5*(2*Ea2pp + 3*Ee1p - 5*Ee2p))/14, k == 2 && m == 0}, {(3*(3*Ea2pp + Ee1p + 3*Ee2p - 7*Ee2pp))/7, k == 4 && m == 0}, {0, k != 6 || m != 0}}, (13*(10*Ea2pp - 15*Ee1p - Ee2p + 6*Ee2pp))/70]

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

Akm = {{0, 0, (1/7)*(Ea2pp + (2)*(Ee1p + Ee2p + Ee2pp))} , 
       {2, 0, (5/14)*((2)*(Ea2pp) + (3)*(Ee1p) + (-5)*(Ee2p))} , 
       {4, 0, (3/7)*((3)*(Ea2pp) + Ee1p + (3)*(Ee2p) + (-7)*(Ee2pp))} , 
       {6, 0, (13/70)*((10)*(Ea2pp) + (-15)*(Ee1p) + (-1)*(Ee2p) + (6)*(Ee2pp))} }

</code>

###

</hidden>
<hidden **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)}} $|$ \text{Ee2p} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \text{Ee2pp} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \text{Ee1p} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea2pp} $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{1}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1p} $|$ 0 $|$ 0 $|
^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2pp} $|$ 0 $|
^$ {Y_{3}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2p} $|


###

</hidden>
<hidden **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{Ee2p} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{\text{xyz}} $|$ 0 $|$ \text{Ee2pp} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Ee1p} $|$ 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{Ee1p} $|$ 0 $|$ 0 $|
^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2pp} $|$ 0 $|
^$ f_{x\left(x^2-3y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2p} $|


###

</hidden>
<hidden **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}} $|


###

</hidden>
<hidden **Irriducible representations and their onsite energy** >

###

^ ^$$\text{Ee2p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$$ | ::: |
^ ^$$\text{Ee2pp}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
^ ^$$\text{Ee1p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$$ | ::: |
^ ^$$\text{Ea2pp}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
^ ^$$\text{Ee1p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$$ | ::: |
^ ^$$\text{Ee2pp}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |
^ ^$$\text{Ee2p}$$ | {{:physics_chemistry:pointgroup:d5h_zy_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$$ | ::: |


###

</hidden>
===== Coupling between two shells =====



###

Click on one of the subsections to expand it or <hiddenSwitch expand all> 

###

==== Potential for s-d orbital mixing ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

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

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

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

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

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

</code>

###

</hidden>
<hidden **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 $|


###

</hidden>
<hidden **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 $|


###

</hidden>
==== Potential for p-f orbital mixing ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

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

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

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

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

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

</code>

###

</hidden>
<hidden **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 $|


###

</hidden>
<hidden **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 $|


###

</hidden>
==== Potential for d-f orbital mixing ====

<hidden **Potential parameterized with onsite energies of irriducible representations** >

###

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

###

</hidden>
<hidden **Input format suitable for Mathematica (Quanty.nb)** >

###

<code Quanty Akm_D5h_Zy.Quanty.nb>

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

</code>

###

</hidden><hidden **Input format suitable for Quanty** >

###

<code Quanty Akm_D5h_Zy.Quanty>

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

</code>

###

</hidden>
<hidden **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 $|$ 0 $|$ 0 $|$ -\frac{5}{11} i \sqrt{\frac{2}{3}} B(5,5) $|
^$ {Y_{-1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ {Y_{2}^{(2)}} $|$ -\frac{5}{11} i \sqrt{\frac{2}{3}} B(5,5) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|


###

</hidden>
<hidden **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 $|$ 0 $|$ 0 $|$ \frac{5}{11} \sqrt{\frac{2}{3}} B(5,5) $|
^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
^$ d_{x^2-y^2} $|$ \frac{5}{11} \sqrt{\frac{2}{3}} B(5,5) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|


###

</hidden>

===== Table of several point groups =====

###

[[physics_chemistry:point_groups|Return to Main page on Point Groups]]

###

###

^Nonaxial groups      | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> | | | | |
^C<sub>n</sub> groups | [[physics_chemistry:point_groups:c2|C]]<sub>[[physics_chemistry:point_groups:c2|2]]</sub> | [[physics_chemistry:point_groups:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> | 
^D<sub>n</sub> groups | [[physics_chemistry:point_groups:d2|D]]<sub>[[physics_chemistry:point_groups:d2|2]]</sub> | [[physics_chemistry:point_groups:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub> | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub> | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub> | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub> | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub> | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub> | 
^C<sub>nv</sub> groups | [[physics_chemistry:point_groups:c2v|C]]<sub>[[physics_chemistry:point_groups:c2v|2v]]</sub> | [[physics_chemistry:point_groups:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> | 
^C<sub>nh</sub> groups | [[physics_chemistry:point_groups:c2h|C]]<sub>[[physics_chemistry:point_groups:c2h|2h]]</sub> | [[physics_chemistry:point_groups:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> | | | 
^D<sub>nh</sub> groups | [[physics_chemistry:point_groups:d2h|D]]<sub>[[physics_chemistry:point_groups:d2h|2h]]</sub> | [[physics_chemistry:point_groups:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> | 
^D<sub>nd</sub> groups | [[physics_chemistry:point_groups:d2d|D]]<sub>[[physics_chemistry:point_groups:d2d|2d]]</sub> | [[physics_chemistry:point_groups:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> | 
^S<sub>n</sub> groups | [[physics_chemistry:point_groups:S2|S]]<sub>[[physics_chemistry:point_groups:S2|2]]</sub> | [[physics_chemistry:point_groups:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> |  | 
^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> | 
^Linear groups      | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | |

###