Table of Contents

Orientation xyz

Symmetry Operations

In the Th Point Group, with orientation xyz there are the following symmetry operations

Operator Orientation
$\text{E}$ $\{0,0,0\}$ ,
$C_3$ $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ ,
$C_2$ $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ ,
$\text{i}$ $\{0,0,0\}$ ,
$S_6$ $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ ,
$\sigma _h$ $\{1,0,0\}$ , $\{0,1,0\}$ , $\{0,0,1\}$ ,

Different Settings

Character Table

$ $ $ \text{E} \,{\text{(1)}} $ $ C_3 \,{\text{(8)}} $ $ C_2 \,{\text{(3)}} $ $ \text{i} \,{\text{(1)}} $ $ S_6 \,{\text{(8)}} $ $ \sigma_h \,{\text{(3)}} $
$ A_g $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ E_g $ $ 2 $ $ -1 $ $ 2 $ $ 2 $ $ -1 $ $ 2 $
$ T_g $ $ 3 $ $ 0 $ $ -1 $ $ 3 $ $ 0 $ $ -1 $
$ A_u $ $ 1 $ $ 1 $ $ 1 $ $ -1 $ $ -1 $ $ -1 $
$ E_u $ $ 2 $ $ -1 $ $ 2 $ $ -2 $ $ 1 $ $ -2 $
$ T_u $ $ 3 $ $ 0 $ $ -1 $ $ -3 $ $ 0 $ $ 1 $

Product Table

$ $ $ A_g $ $ E_g $ $ T_g $ $ A_u $ $ E_u $ $ T_u $
$ A_g $ $ A_g $ $ E_g $ $ T_g $ $ A_u $ $ E_u $ $ T_u $
$ E_g $ $ E_g $ $ 2 A_g+E_g $ $ 2 T_g $ $ E_u $ $ 2 A_u+E_u $ $ 2 T_u $
$ T_g $ $ T_g $ $ 2 T_g $ $ A_g+E_g+2 T_g $ $ T_u $ $ 2 T_u $ $ A_u+E_u+2 T_u $
$ A_u $ $ A_u $ $ E_u $ $ T_u $ $ A_g $ $ E_g $ $ T_g $
$ E_u $ $ E_u $ $ 2 A_u+E_u $ $ 2 T_u $ $ E_g $ $ 2 A_g+E_g $ $ 2 T_g $
$ T_u $ $ T_u $ $ 2 T_u $ $ A_u+E_u+2 T_u $ $ T_g $ $ 2 T_g $ $ A_g+E_g+2 T_g $

Sub Groups with compatible settings

Super Groups with compatible settings

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 Th Point group with orientation xyz the form of the expansion coefficients is:

Expansion

$$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ A(4,0) & k=4\land m=0 \\ -\sqrt{\frac{5}{11}} A(6,2) & k=6\land (m=-6\lor m=6) \\ -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ A(6,2) & k=6\land (m=-2\lor m=2) \\ A(6,0) & k=6\land m=0 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Akm_Th_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[5/11]*A[6, 2]), k == 6 && (m == -6 || m == 6)}, {-(Sqrt[7/2]*A[6, 0]), k == 6 && (m == -4 || m == 4)}, {A[6, 2], k == 6 && (m == -2 || m == 2)}, {A[6, 0], k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Th_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,-2, A(6,2)} , 
       {6, 2, A(6,2)} , 
       {6,-4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6, 4, (-1)*((sqrt(7/2))*(A(6,0)))} , 
       {6,-6, (-1)*((sqrt(5/11))*(A(6,2)))} , 
       {6, 6, (-1)*((sqrt(5/11))*(A(6,2)))} }

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) $$ 0 $$ -\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $$ 0 $$ \frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $
$ {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 $$ \frac{20}{429} \sqrt{14} \text{Aff}(6,2) $$ 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 }$$\color{darkred}{ 0 }$$ -\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $$ 0 $$ -\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $$ 0 $$ \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 $$ \frac{20}{429} \sqrt{14} \text{Aff}(6,2) $$ 0 $$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $$ 0 $$ \frac{20}{429} \sqrt{14} \text{Aff}(6,2) $$ 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) $$ 0 $$ -\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $$ 0 $$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $$ 0 $$ -\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $
$ {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 $$ \frac{20}{429} \sqrt{14} \text{Aff}(6,2) $$ 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 }$$\color{darkred}{ 0 }$$ \frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,2) $$ 0 $$ \frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0) $$ 0 $$ -\frac{10}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $$ \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 $$ 0 $$ \frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $$ 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 $$ \frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 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 $$ 0 $$ \frac{40}{429} \sqrt{7} \text{Aff}(6,2) $
$ 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 $$ \frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 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 $$ 0 $$ \frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 0 $$ 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 $$ \frac{40}{429} \sqrt{7} \text{Aff}(6,2) $$ 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

Potential for s orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Th_xyz.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Eag, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Th_xyz.Quanty
Akm = {{0, 0, Eag} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{0}^{(0)}} $
$ {Y_{0}^{(0)}} $$ \text{Eag} $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ \text{s} $
$ \text{s} $$ \text{Eag} $

Rotation matrix used

Rotation matrix used

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

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

$$\text{Eag}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2 \sqrt{\pi }}$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2 \sqrt{\pi }}$$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

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

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Th_xyz.Quanty.nb
Akm[k_,m_]:=Piecewise[{{Etu, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Th_xyz.Quanty
Akm = {{0, 0, Etu} }

The Hamiltonian on a basis of spherical Harmonics

The Hamiltonian on a basis of spherical Harmonics

$ $ $ {Y_{-1}^{(1)}} $ $ {Y_{0}^{(1)}} $ $ {Y_{1}^{(1)}} $
$ {Y_{-1}^{(1)}} $$ \text{Etu} $$ 0 $$ 0 $
$ {Y_{0}^{(1)}} $$ 0 $$ \text{Etu} $$ 0 $
$ {Y_{1}^{(1)}} $$ 0 $$ 0 $$ \text{Etu} $

The Hamiltonian on a basis of symmetric functions

The Hamiltonian on a basis of symmetric functions

$ $ $ p_x $ $ p_y $ $ p_z $
$ p_x $$ \text{Etu} $$ 0 $$ 0 $
$ p_y $$ 0 $$ \text{Etu} $$ 0 $
$ p_z $$ 0 $$ 0 $$ \text{Etu} $

Rotation matrix used

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 $

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

$$\text{Etu}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$
$$\text{Etu}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$
$$\text{Etu}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{2 \text{Eeg}}{5}+\frac{3 \text{Etg}}{5} & k=0\land m=0 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eeg}-\text{Etg}) & k=4\land (m=-4\lor m=4) \\ \frac{21 (\text{Eeg}-\text{Etg})}{10} & k=4\land m=0 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Th_xyz.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(2*Eeg)/5 + (3*Etg)/5, k == 0 && m == 0}, {(3*Sqrt[7/10]*(Eeg - Etg))/2, k == 4 && (m == -4 || m == 4)}, {(21*(Eeg - Etg))/10, k == 4 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Th_xyz.Quanty
Akm = {{0, 0, (2/5)*(Eeg) + (3/5)*(Etg)} , 
       {4, 0, (21/10)*(Eeg + (-1)*(Etg))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Eeg + (-1)*(Etg)))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Eeg + (-1)*(Etg)))} }

The Hamiltonian on a basis of spherical Harmonics

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)}} $$ \frac{\text{Eeg}+\text{Etg}}{2} $$ 0 $$ 0 $$ 0 $$ \frac{\text{Eeg}-\text{Etg}}{2} $
$ {Y_{-1}^{(2)}} $$ 0 $$ \text{Etg} $$ 0 $$ 0 $$ 0 $
$ {Y_{0}^{(2)}} $$ 0 $$ 0 $$ \text{Eeg} $$ 0 $$ 0 $
$ {Y_{1}^{(2)}} $$ 0 $$ 0 $$ 0 $$ \text{Etg} $$ 0 $
$ {Y_{2}^{(2)}} $$ \frac{\text{Eeg}-\text{Etg}}{2} $$ 0 $$ 0 $$ 0 $$ \frac{\text{Eeg}+\text{Etg}}{2} $

The Hamiltonian on a basis of symmetric functions

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} $$ \text{Eeg} $$ 0 $$ 0 $$ 0 $$ 0 $
$ d_{3z^2-r^2} $$ 0 $$ \text{Eeg} $$ 0 $$ 0 $$ 0 $
$ d_{\text{yz}} $$ 0 $$ 0 $$ \text{Etg} $$ 0 $$ 0 $
$ d_{\text{xz}} $$ 0 $$ 0 $$ 0 $$ \text{Etg} $$ 0 $
$ d_{\text{xy}} $$ 0 $$ 0 $$ 0 $$ 0 $$ \text{Etg} $

Rotation matrix used

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}} $

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

$$\text{Eeg}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$
$$\text{Eeg}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$
$$\text{Etg}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$
$$\text{Etg}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$
$$\text{Etg}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Eau}+3 (\text{Etu1}+\text{Etu2})) & k=0\land m=0 \\ -\frac{3}{4} \sqrt{\frac{5}{14}} (2 \text{Eau}-3 \text{Etu1}+\text{Etu2}) & k=4\land (m=-4\lor m=4) \\ -\frac{3}{4} (2 \text{Eau}-3 \text{Etu1}+\text{Etu2}) & k=4\land m=0 \\ -\frac{39}{8} \sqrt{\frac{11}{35}} \text{Mtu} & k=6\land (m=-6\lor m=6) \\ -\frac{39 (4 \text{Eau}+5 \text{Etu1}-9 \text{Etu2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ \frac{429 \text{Mtu}}{40 \sqrt{7}} & k=6\land (m=-2\lor m=2) \\ \frac{39}{280} (4 \text{Eau}+5 \text{Etu1}-9 \text{Etu2}) & k=6\land m=0 \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Th_xyz.Quanty.nb
Akm[k_,m_]:=Piecewise[{{(Eau + 3*(Etu1 + Etu2))/7, k == 0 && m == 0}, {(-3*Sqrt[5/14]*(2*Eau - 3*Etu1 + Etu2))/4, k == 4 && (m == -4 || m == 4)}, {(-3*(2*Eau - 3*Etu1 + Etu2))/4, k == 4 && m == 0}, {(-39*Sqrt[11/35]*Mtu)/8, k == 6 && (m == -6 || m == 6)}, {(-39*(4*Eau + 5*Etu1 - 9*Etu2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(429*Mtu)/(40*Sqrt[7]), k == 6 && (m == -2 || m == 2)}, {(39*(4*Eau + 5*Etu1 - 9*Etu2))/280, k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Th_xyz.Quanty
Akm = {{0, 0, (1/7)*(Eau + (3)*(Etu1 + Etu2))} , 
       {4, 0, (-3/4)*((2)*(Eau) + (-3)*(Etu1) + Etu2)} , 
       {4,-4, (-3/4)*((sqrt(5/14))*((2)*(Eau) + (-3)*(Etu1) + Etu2))} , 
       {4, 4, (-3/4)*((sqrt(5/14))*((2)*(Eau) + (-3)*(Etu1) + Etu2))} , 
       {6, 0, (39/280)*((4)*(Eau) + (5)*(Etu1) + (-9)*(Etu2))} , 
       {6,-2, (429/40)*((1/(sqrt(7)))*(Mtu))} , 
       {6, 2, (429/40)*((1/(sqrt(7)))*(Mtu))} , 
       {6,-4, (-39/40)*((1/(sqrt(14)))*((4)*(Eau) + (5)*(Etu1) + (-9)*(Etu2)))} , 
       {6, 4, (-39/40)*((1/(sqrt(14)))*((4)*(Eau) + (5)*(Etu1) + (-9)*(Etu2)))} , 
       {6,-6, (-39/8)*((sqrt(11/35))*(Mtu))} , 
       {6, 6, (-39/8)*((sqrt(11/35))*(Mtu))} }

The Hamiltonian on a basis of spherical Harmonics

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)}} $$ \frac{1}{8} (5 \text{Etu1}+3 \text{Etu2}) $$ 0 $$ -\frac{\text{Mtu}}{4} $$ 0 $$ \frac{1}{8} \sqrt{15} (\text{Etu1}-\text{Etu2}) $$ 0 $$ \frac{\sqrt{15} \text{Mtu}}{4} $
$ {Y_{-2}^{(3)}} $$ 0 $$ \frac{\text{Eau}+\text{Etu2}}{2} $$ 0 $$ \frac{\text{Mtu}}{\sqrt{2}} $$ 0 $$ \frac{\text{Etu2}-\text{Eau}}{2} $$ 0 $
$ {Y_{-1}^{(3)}} $$ -\frac{\text{Mtu}}{4} $$ 0 $$ \frac{1}{8} (3 \text{Etu1}+5 \text{Etu2}) $$ 0 $$ -\frac{\sqrt{15} \text{Mtu}}{4} $$ 0 $$ \frac{1}{8} \sqrt{15} (\text{Etu1}-\text{Etu2}) $
$ {Y_{0}^{(3)}} $$ 0 $$ \frac{\text{Mtu}}{\sqrt{2}} $$ 0 $$ \text{Etu1} $$ 0 $$ \frac{\text{Mtu}}{\sqrt{2}} $$ 0 $
$ {Y_{1}^{(3)}} $$ \frac{1}{8} \sqrt{15} (\text{Etu1}-\text{Etu2}) $$ 0 $$ -\frac{\sqrt{15} \text{Mtu}}{4} $$ 0 $$ \frac{1}{8} (3 \text{Etu1}+5 \text{Etu2}) $$ 0 $$ -\frac{\text{Mtu}}{4} $
$ {Y_{2}^{(3)}} $$ 0 $$ \frac{\text{Etu2}-\text{Eau}}{2} $$ 0 $$ \frac{\text{Mtu}}{\sqrt{2}} $$ 0 $$ \frac{\text{Eau}+\text{Etu2}}{2} $$ 0 $
$ {Y_{3}^{(3)}} $$ \frac{\sqrt{15} \text{Mtu}}{4} $$ 0 $$ \frac{1}{8} \sqrt{15} (\text{Etu1}-\text{Etu2}) $$ 0 $$ -\frac{\text{Mtu}}{4} $$ 0 $$ \frac{1}{8} (5 \text{Etu1}+3 \text{Etu2}) $

The Hamiltonian on a basis of symmetric functions

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}} $$ \text{Eau} $$ 0 $$ 0 $$ 0 $$ 0 $$ 0 $$ 0 $
$ f_{x\left(5x^2-r^2\right)} $$ 0 $$ \text{Etu1} $$ 0 $$ 0 $$ \text{Mtu} $$ 0 $$ 0 $
$ f_{y\left(5y^2-r^2\right)} $$ 0 $$ 0 $$ \text{Etu1} $$ 0 $$ 0 $$ \text{Mtu} $$ 0 $
$ f_{z\left(5z^2-r^2\right)} $$ 0 $$ 0 $$ 0 $$ \text{Etu1} $$ 0 $$ 0 $$ \text{Mtu} $
$ f_{x\left(y^2-z^2\right)} $$ 0 $$ \text{Mtu} $$ 0 $$ 0 $$ \text{Etu2} $$ 0 $$ 0 $
$ f_{y\left(z^2-x^2\right)} $$ 0 $$ 0 $$ \text{Mtu} $$ 0 $$ 0 $$ \text{Etu2} $$ 0 $
$ f_{z\left(x^2-y^2\right)} $$ 0 $$ 0 $$ 0 $$ \text{Mtu} $$ 0 $$ 0 $$ \text{Etu2} $

Rotation matrix used

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 $

Irriducible representations and their onsite energy

Irriducible representations and their onsite energy

$$\text{Eau}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$
$$\text{Etu1}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$
$$\text{Etu1}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$
$$\text{Etu1}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$
$$\text{Etu2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$
$$\text{Etu2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$
$$\text{Etu2}$$
$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$
$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ $$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

Potential parameterized with onsite energies of irriducible representations

$$A_{k,m} = \begin{cases} 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ A(4,0) & \text{True} \end{cases}$$

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Mathematica (Quanty.nb)

Akm_Th_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]]

Input format suitable for Quanty

Input format suitable for Quanty

Akm_Th_xyz.Quanty
Akm = {{4, 0, A(4,0)} , 
       {4,-4, (sqrt(5/14))*(A(4,0))} , 
       {4, 4, (sqrt(5/14))*(A(4,0))} }

The Hamiltonian on a basis of spherical Harmonics

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 $

The Hamiltonian on a basis of symmetric functions

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 $

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups C1 Cs Ci
Cn groups C2 C3 C4 C5 C6 C7 C8
Dn groups D2 D3 D4 D5 D6 D7 D8
Cnv groups C2v C3v C4v C5v C6v C7v C8v
Cnh groups C2h C3h C4h C5h C6h
Dnh groups D2h D3h D4h D5h D6h D7h D8h
Dnd groups D2d D3d D4d D5d D6d D7d D8d
Sn groups S2 S4 S6 S8 S10 S12
Cubic groups T Th Td O Oh I Ih
Linear groups C$\infty$v D$\infty$h

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