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physics_chemistry:point_groups:o:orientation_xyz [2018/03/21 18:46] – created Stefano Agrestiniphysics_chemistry:point_groups:o:orientation_xyz [2018/04/06 08:55] (current) – Maurits W. Haverkort
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 +~~CLOSETOC~~
 +
 ====== Orientation XYZ ====== ====== Orientation XYZ ======
 +
 +===== Symmetry Operations =====
  
 ### ###
-alligned paragraph text+ 
 +In the O Point Group, with orientation XYZ there are the following symmetry operations 
 ### ###
  
-===== Example =====+### 
 + 
 +{{:physics_chemistry:pointgroup:o_xyz.png}}
  
 ### ###
-description text+
 ### ###
  
-==== Input ==== +^ Operator ^ Orientation ^ 
-<code Quanty Example.Quanty> +^ $\text{E}$ | $\{0,0,0\}$ , | 
--- some example code+^ $C_3$ | $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ , | 
 +^ $C_2$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ , | 
 +^ $C_4$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ , | 
 +^ $C_2$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , | 
 + 
 +### 
 + 
 +===== Different Settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:point_groups:o:orientation_xyz|Point Group O with orientation XYZ]] 
 + 
 +### 
 + 
 +===== Character Table ===== 
 + 
 +### 
 + 
 +|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_3 \,{\text{(8)}} $  ^  $ C_2 \,{\text{(6)}} $  ^  $ C_4 \,{\text{(6)}} $  ^  $ C_2 \,{\text{(3)}} $  ^ 
 +^ $ A_1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ | 
 +^ $ A_2 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ | 
 +^ $ \text{E} $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 0 $ |  $ 2 $ | 
 +^ $ T_1 $ |  $ 3 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ | 
 +^ $ T_2 $ |  $ 3 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ | 
 + 
 +### 
 + 
 +===== Product Table ===== 
 + 
 +### 
 + 
 +|  $  $  ^  $ A_1 $  ^  $ A_2 $  ^  $ \text{E} $  ^  $ T_1 $  ^  $ T_2 $  ^ 
 +^ $ A_1 $  | $ A_1 $  | $ A_2 $  | $ \text{E} $  | $ T_1 $  | $ T_2 $  | 
 +^ $ A_2 $  | $ A_2 $  | $ A_1 $  | $ \text{E} $  | $ T_2 $  | $ T_1 $  | 
 +^ $ \text{E} $  | $ \text{E} $  | $ \text{E} $  | $ \text{E}+A_1+A_2 $  | $ T_1+T_2 $  | $ T_1+T_2 $  | 
 +^ $ T_1 $  | $ T_1 $  | $ T_2 $  | $ T_1+T_2 $  | $ \text{E}+A_1+T_1+T_2 $  | $ \text{E}+A_2+T_1+T_2 $  | 
 +^ $ T_2 $  | $ T_2 $  | $ T_1 $  | $ T_1+T_2 $  | $ \text{E}+A_2+T_1+T_2 $  | $ \text{E}+A_1+T_1+T_2 $  | 
 + 
 +### 
 + 
 +===== Sub Groups with compatible settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] 
 +  * [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]] 
 +  * [[physics_chemistry:point_groups:c2:orientation_y|Point Group C2 with orientation Y]] 
 +  * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]] 
 +  * [[physics_chemistry:point_groups:c4:orientation_x|Point Group C4 with orientation X]] 
 +  * [[physics_chemistry:point_groups:c4:orientation_y|Point Group C4 with orientation Y]] 
 +  * [[physics_chemistry:point_groups:c4:orientation_z|Point Group C4 with orientation Z]] 
 +  * [[physics_chemistry:point_groups:d2:orientation_xyz|Point Group D2 with orientation XYZ]] 
 +  * [[physics_chemistry:point_groups:d4:orientation_zxy|Point Group D4 with orientation Zxy]] 
 +  * [[physics_chemistry:point_groups:t:orientation_xyz|Point Group T with orientation xyz]] 
 + 
 +### 
 + 
 +===== Super Groups with compatible settings ===== 
 + 
 +### 
 + 
 +  * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] 
 + 
 +### 
 + 
 +===== 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 O Point group with orientation XYZ the form of the expansion coefficients is: 
 + 
 +### 
 + 
 +==== Expansion ==== 
 + 
 +### 
 + 
 + $$A_{k,m} = \begin{cases} 
 + A(0,0) & k=0\land m=0 \\ 
 + \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ 
 + A(4,0) & k=4\land m=0 \\ 
 + -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ 
 + A(6,0) & k=6\land m=0 
 +\end{cases}$$ 
 + 
 +### 
 + 
 +==== Input format suitable for Mathematica (Quanty.nb) ==== 
 + 
 +### 
 + 
 +<code Quanty Akm_O_XYZ.Quanty.nb> 
 + 
 +Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {A[6, 0], k == 6 && m == 0}}, 0] 
 </code> </code>
  
-==== Result ==== +###
-<WRAP center box 100%> +
-text produced as output +
-</WRAP>+
  
-===== Table of contents ===== +==== Input format suitable for Quanty ====
-{{indexmenu>.#1}}+
  
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty>
 +
 +Akm = {{0, 0, A(0,0)} , 
 +       {4, 0, A(4,0)} , 
 +       {4,-4, (sqrt(5/14))*(A(4,0))} , 
 +       {4, 4, (sqrt(5/14))*(A(4,0))} , 
 +       {6, 0, A(6,0)} , 
 +       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
 +       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} }
 +
 +</code>
 +
 +###
 +
 +==== 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 $|$ 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}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|
 +^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 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) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ 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{1}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{21} \text{Add}(4,0) $|$\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 $|$ \text{Add}(0,0)-\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)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\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{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 }$|$ \frac{5}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\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 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \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 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $|$ 0 $|
 +^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 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}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|
 +^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 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{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{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 }$|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\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 $|$ \frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0) $|$ 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 }$|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\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_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 $|
 +^$ 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 $|
 +^$ 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 }$|
 +^$ 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{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_{\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_{\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 }$|
 +^$ 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_{x\left(5x^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 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
 +^$ f_{y\left(5y^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 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
 +^$ f_{z\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 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
 +^$ f_{y\left(z^2-x^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{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
 +^$ 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 $|
 +
 +
 +###
 +
 +==== One particle coupling on a basis of symmetry adapted functions ====
 +
 +###
 +
 +After rotation we find
 +
 +###
 +
 +
 +
 +###
 +
 +|  $  $  ^  $ \text{s} $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
 +^$ \text{s} $|$ \text{Ass}(0,0) $|$\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_x $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{2}{7} \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}{ 0 }$|
 +^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \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_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{4}{21} \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{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_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 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 }$|$ \text{Aff}(0,0)-\frac{4}{11} \text{Aff}(4,0)+\frac{80}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 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 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x\left(y^2-z^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 $|$ \text{Aff}(0,0)-\frac{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|
 +^$ f_{y\left(z^2-x^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{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0) $|$ 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 $|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{33} \text{Aff}(4,0)-\frac{60}{143} \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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \text{Ea1} & k=0\land m=0 \\
 + 0 & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{Ea1, k == 0 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty>
 +
 +Akm = {{0, 0, Ea1} }
 +
 +</code>
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
 +^$ {Y_{0}^{(0)}} $|$ \text{Ea1} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ \text{s} $  ^
 +^$ \text{s} $|$ \text{Ea1} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
 +^$ \text{s} $|$ 1 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea1}$$ | {{:physics_chemistry:pointgroup:o_xyz_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** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \text{Et1} & k=0\land m=0 \\
 + 0 & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{Et1, k == 0 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty>
 +
 +Akm = {{0, 0, Et1} }
 +
 +</code>
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of spherical Harmonics** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
 +^$ {Y_{-1}^{(1)}} $|$ \text{Et1} $|$ 0 $|$ 0 $|
 +^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Et1} $|$ 0 $|
 +^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Et1} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ p_x $  ^  $ p_y $  ^  $ p_z $  ^
 +^$ p_x $|$ \text{Et1} $|$ 0 $|$ 0 $|
 +^$ p_y $|$ 0 $|$ \text{Et1} $|$ 0 $|
 +^$ p_z $|$ 0 $|$ 0 $|$ \text{Et1} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-1}^{(1)}} $  ^  $ {Y_{0}^{(1)}} $  ^  $ {Y_{1}^{(1)}} $  ^
 +^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|
 +^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|
 +^$ p_z $|$ 0 $|$ 1 $|$ 0 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Et1}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_1_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
 +^ ^$$\text{Et1}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_1_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
 +^ ^$$\text{Et1}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_1_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
 +
 +
 +###
 +
 +</hidden>
 +==== Potential for d orbitals ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \frac{2 \text{Ee}}{5}+\frac{3 \text{Et2}}{5} & k=0\land m=0 \\
 + \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ee}-\text{Et2}) & k=4\land (m=-4\lor m=4) \\
 + \frac{21 (\text{Ee}-\text{Et2})}{10} & k=4\land m=0
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{(2*Ee)/5 + (3*Et2)/5, k == 0 && m == 0}, {(3*Sqrt[7/10]*(Ee - Et2))/2, k == 4 && (m == -4 || m == 4)}, {(21*(Ee - Et2))/10, k == 4 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty>
 +
 +Akm = {{0, 0, (2/5)*(Ee) + (3/5)*(Et2)} , 
 +       {4, 0, (21/10)*(Ee + (-1)*(Et2))} , 
 +       {4,-4, (3/2)*((sqrt(7/10))*(Ee + (-1)*(Et2)))} , 
 +       {4, 4, (3/2)*((sqrt(7/10))*(Ee + (-1)*(Et2)))} }
 +
 +</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)}} $|$ \frac{\text{Ee}+\text{Et2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ee}-\text{Et2}}{2} $|
 +^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ee} $|$ 0 $|$ 0 $|
 +^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|
 +^$ {Y_{2}^{(2)}} $|$ \frac{\text{Ee}-\text{Et2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ee}+\text{Et2}}{2} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ d_{x^2-y^2} $  ^  $ d_{3z^2-r^2} $  ^  $ d_{\text{yz}} $  ^  $ d_{\text{xz}} $  ^  $ d_{\text{xy}} $  ^
 +^$ d_{x^2-y^2} $|$ \text{Ee} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{3z^2-r^2} $|$ 0 $|$ \text{Ee} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|
 +^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|
 +^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Rotation matrix used** >
 +
 +###
 +
 +|  $  $  ^  $ {Y_{-2}^{(2)}} $  ^  $ {Y_{-1}^{(2)}} $  ^  $ {Y_{0}^{(2)}} $  ^  $ {Y_{1}^{(2)}} $  ^  $ {Y_{2}^{(2)}} $  ^
 +^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|
 +^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|
 +^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|
 +^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|
 +^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_2_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: |
 +^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_2_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Et2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_2_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: |
 +^ ^$$\text{Et2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_2_4.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: |
 +^ ^$$\text{Et2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_2_5.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: |
 +
 +
 +###
 +
 +</hidden>
 +==== Potential for f orbitals ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + \frac{1}{7} (\text{Ea2}+3 (\text{Et1}+\text{Et2})) & k=0\land m=0 \\
 + -\frac{3}{4} \sqrt{\frac{5}{14}} (2 \text{Ea2}-3 \text{Et1}+\text{Et2}) & k=4\land (m=-4\lor m=4) \\
 + -\frac{3}{4} (2 \text{Ea2}-3 \text{Et1}+\text{Et2}) & k=4\land m=0 \\
 + -\frac{39 (4 \text{Ea2}+5 \text{Et1}-9 \text{Et2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\
 + \frac{39}{280} (4 \text{Ea2}+5 \text{Et1}-9 \text{Et2}) & k=6\land m=0
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{(Ea2 + 3*(Et1 + Et2))/7, k == 0 && m == 0}, {(-3*Sqrt[5/14]*(2*Ea2 - 3*Et1 + Et2))/4, k == 4 && (m == -4 || m == 4)}, {(-3*(2*Ea2 - 3*Et1 + Et2))/4, k == 4 && m == 0}, {(-39*(4*Ea2 + 5*Et1 - 9*Et2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(39*(4*Ea2 + 5*Et1 - 9*Et2))/280, k == 6 && m == 0}}, 0]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty>
 +
 +Akm = {{0, 0, (1/7)*(Ea2 + (3)*(Et1 + Et2))} , 
 +       {4, 0, (-3/4)*((2)*(Ea2) + (-3)*(Et1) + Et2)} , 
 +       {4,-4, (-3/4)*((sqrt(5/14))*((2)*(Ea2) + (-3)*(Et1) + Et2))} , 
 +       {4, 4, (-3/4)*((sqrt(5/14))*((2)*(Ea2) + (-3)*(Et1) + Et2))} , 
 +       {6, 0, (39/280)*((4)*(Ea2) + (5)*(Et1) + (-9)*(Et2))} , 
 +       {6,-4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2) + (5)*(Et1) + (-9)*(Et2)))} , 
 +       {6, 4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2) + (5)*(Et1) + (-9)*(Et2)))} }
 +
 +</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)}} $|$ \frac{1}{8} (5 \text{Et1}+3 \text{Et2}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et1}-\text{Et2}) $|$ 0 $|$ 0 $|
 +^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Ea2}+\text{Et2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Et2}-\text{Ea2}}{2} $|$ 0 $|
 +^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} (3 \text{Et1}+5 \text{Et2}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et1}-\text{Et2}) $|
 +^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \sqrt{15} (\text{Et1}-\text{Et2}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} (3 \text{Et1}+5 \text{Et2}) $|$ 0 $|$ 0 $|
 +^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{\text{Et2}-\text{Ea2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ea2}+\text{Et2}}{2} $|$ 0 $|
 +^$ {Y_{3}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et1}-\text{Et2}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} (5 \text{Et1}+3 \text{Et2}) $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
 +^$ f_{\text{xyz}} $|$ \text{Ea2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Et1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Et1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|
 +^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|
 +^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|
 +
 +
 +###
 +
 +</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_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $|
 +^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|
 +^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $|
 +^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|
 +^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|
 +^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **Irriducible representations and their onsite energy** >
 +
 +###
 +
 +^ ^$$\text{Ea2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_1.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: |
 +^ ^$$\text{Et1}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_2.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: |
 +^ ^$$\text{Et1}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_3.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: |
 +^ ^$$\text{Et1}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_4.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: |
 +^ ^$$\text{Et2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_5.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Et2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_6.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: |
 +^ ^$$\text{Et2}$$ | {{:physics_chemistry:pointgroup:o_xyz_orb_3_7.png?150}} |
 +|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: |
 +|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: |
 +
 +
 +###
 +
 +</hidden>
 +===== Coupling between two shells =====
 +
 +
 +
 +###
 +
 +Click on one of the subsections to expand it or <hiddenSwitch expand all> 
 +
 +###
 +
 +==== Potential for p-f orbital mixing ====
 +
 +<hidden **Potential parameterized with onsite energies of irriducible representations** >
 +
 +###
 +
 + $$A_{k,m} = \begin{cases}
 + 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\
 + \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\
 + A(4,0) & \text{True}
 +\end{cases}$$
 +
 +###
 +
 +</hidden>
 +<hidden **Input format suitable for Mathematica (Quanty.nb)** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty.nb>
 +
 +Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -4 && m != 0 && m != 4)}, {Sqrt[5/14]*A[4, 0], k == 4 && (m == -4 || m == 4)}}, A[4, 0]]
 +
 +</code>
 +
 +###
 +
 +</hidden><hidden **Input format suitable for Quanty** >
 +
 +###
 +
 +<code Quanty Akm_O_XYZ.Quanty>
 +
 +Akm = {{4, 0, A(4,0)} , 
 +       {4,-4, (sqrt(5/14))*(A(4,0))} , 
 +       {4, 4, (sqrt(5/14))*(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}{3} \sqrt{\frac{2}{7}} A(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} A(4,0) $|
 +^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{4 A(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|
 +^$ {Y_{1}^{(1)}} $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} A(4,0) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} A(4,0) $|$ 0 $|$ 0 $|
 +
 +
 +###
 +
 +</hidden>
 +<hidden **The Hamiltonian on a basis of symmetric functions** >
 +
 +###
 +
 +|  $  $  ^  $ f_{\text{xyz}} $  ^  $ f_{x\left(5x^2-r^2\right)} $  ^  $ f_{y\left(5y^2-r^2\right)} $  ^  $ f_{z\left(5z^2-r^2\right)} $  ^  $ f_{x\left(y^2-z^2\right)} $  ^  $ f_{y\left(z^2-x^2\right)} $  ^  $ f_{z\left(x^2-y^2\right)} $  ^
 +^$ p_x $|$ 0 $|$ \frac{4 A(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_y $|$ 0 $|$ 0 $|$ \frac{4 A(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|
 +^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \frac{4 A(4,0)}{3 \sqrt{21}} $|$ 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> | | | | | |
 +
 +###

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