Orientation XYZ

Symmetry Operations

In the Oh Point Group, with orientation XYZ there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ ,
$C_2$	$\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ ,
$C_4$	$\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ ,
$C_2$	$\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ ,
$S_6$	$\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ ,
$\sigma _h$	$\{1,0,0\}$ , $\{0,1,0\}$ , $\{0,0,1\}$ ,
$\sigma _d$	$\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ ,

Different Settings

Character Table

	$\text{E} \,{\text{(1)}}$	$C_3 \,{\text{(8)}}$	$C_2 \,{\text{(6)}}$	$C_4 \,{\text{(6)}}$	$C_2 \,{\text{(3)}}$	$\text{i} \,{\text{(1)}}$	$S_4 \,{\text{(6)}}$	$S_6 \,{\text{(8)}}$	$\sigma_h \,{\text{(3)}}$	$\sigma_d \,{\text{(6)}}$
$A_{1g}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$A_{2g}$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
$E_g$	$2$	$-1$	$0$	$0$	$2$	$2$	$0$	$-1$	$2$	$0$
$T_{1g}$	$3$	$0$	$-1$	$1$	$-1$	$3$	$1$	$0$	$-1$	$-1$
$T_{2g}$	$3$	$0$	$1$	$-1$	$-1$	$3$	$-1$	$0$	$-1$	$1$
$A_{1u}$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$
$A_{2u}$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
$E_u$	$2$	$-1$	$0$	$0$	$2$	$-2$	$0$	$1$	$-2$	$0$
$T_{1u}$	$3$	$0$	$-1$	$1$	$-1$	$-3$	$-1$	$0$	$1$	$1$
$T_{2u}$	$3$	$0$	$1$	$-1$	$-1$	$-3$	$1$	$0$	$1$	$-1$

Product Table

	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$
$A_{1g}$	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$
$A_{2g}$	$A_{2g}$	$A_{1g}$	$E_g$	$T_{2g}$	$T_{1g}$	$A_{2u}$	$A_{1u}$	$E_u$	$T_{2u}$	$T_{1u}$
$E_g$	$E_g$	$E_g$	$A_{1g}+A_{2g}+E_g$	$T_{1g}+T_{2g}$	$T_{1g}+T_{2g}$	$E_u$	$E_u$	$A_{1u}+A_{2u}+E_u$	$T_{1u}+T_{2u}$	$T_{1u}+T_{2u}$
$T_{1g}$	$T_{1g}$	$T_{2g}$	$T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$T_{1u}$	$T_{2u}$	$T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$
$T_{2g}$	$T_{2g}$	$T_{1g}$	$T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$T_{2u}$	$T_{1u}$	$T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$
$A_{1u}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$
$A_{2u}$	$A_{2u}$	$A_{1u}$	$E_u$	$T_{2u}$	$T_{1u}$	$A_{2g}$	$A_{1g}$	$E_g$	$T_{2g}$	$T_{1g}$
$E_u$	$E_u$	$E_u$	$A_{1u}+A_{2u}+E_u$	$T_{1u}+T_{2u}$	$T_{1u}+T_{2u}$	$E_g$	$E_g$	$A_{1g}+A_{2g}+E_g$	$T_{1g}+T_{2g}$	$T_{1g}+T_{2g}$
$T_{1u}$	$T_{1u}$	$T_{2u}$	$T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$T_{1g}$	$T_{2g}$	$T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$
$T_{2u}$	$T_{2u}$	$T_{1u}$	$T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$T_{2g}$	$T_{1g}$	$T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$

Sub Groups with compatible settings

Super Groups with compatible settings

Point Group Ih 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 Oh 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)

Akm_Oh_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]

Input format suitable for Quanty

Akm_Oh_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)))} }

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

Potential for s orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Ea1g} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_XYZ.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Oh_XYZ.Quanty

Akm = {{0, 0, Ea1g} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{0}^{(0)}}$
${Y_{0}^{(0)}}$	$\text{Ea1g}$

The Hamiltonian on a basis of symmetric functions

	$\text{s}$
$\text{s}$	$\text{Ea1g}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ea1g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Et1u} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_XYZ.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Oh_XYZ.Quanty

Akm = {{0, 0, Et1u} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\text{Et1u}$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$\text{Et1u}$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$\text{Et1u}$

The Hamiltonian on a basis of symmetric functions

	$p_x$	$p_y$	$p_z$
$p_x$	$\text{Et1u}$	$0$	$0$
$p_y$	$0$	$\text{Et1u}$	$0$
$p_z$	$0$	$0$	$\text{Et1u}$

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

	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{2 \text{Eeg}}{5}+\frac{3 \text{Et2g}}{5} & k=0\land m=0 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eeg}-\text{Et2g}) & k=4\land (m=-4\lor m=4) \\ \frac{21 (\text{Eeg}-\text{Et2g})}{10} & k=4\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_XYZ.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(2*Eeg)/5 + (3*Et2g)/5, k == 0 && m == 0}, {(3*Sqrt[7/10]*(Eeg - Et2g))/2, k == 4 && (m == -4 || m == 4)}, {(21*(Eeg - Et2g))/10, k == 4 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Oh_XYZ.Quanty

Akm = {{0, 0, (2/5)*(Eeg) + (3/5)*(Et2g)} , 
       {4, 0, (21/10)*(Eeg + (-1)*(Et2g))} , 
       {4,-4, (3/2)*((sqrt(7/10))*(Eeg + (-1)*(Et2g)))} , 
       {4, 4, (3/2)*((sqrt(7/10))*(Eeg + (-1)*(Et2g)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\frac{\text{Eeg}+\text{Et2g}}{2}$	$0$	$0$	$0$	$\frac{\text{Eeg}-\text{Et2g}}{2}$
${Y_{-1}^{(2)}}$	$0$	$\text{Et2g}$	$0$	$0$	$0$
${Y_{0}^{(2)}}$	$0$	$0$	$\text{Eeg}$	$0$	$0$
${Y_{1}^{(2)}}$	$0$	$0$	$0$	$\text{Et2g}$	$0$
${Y_{2}^{(2)}}$	$\frac{\text{Eeg}-\text{Et2g}}{2}$	$0$	$0$	$0$	$\frac{\text{Eeg}+\text{Et2g}}{2}$

The Hamiltonian on a basis of symmetric functions

	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$
$d_{x^2-y^2}$	$\text{Eeg}$	$0$	$0$	$0$	$0$
$d_{3z^2-r^2}$	$0$	$\text{Eeg}$	$0$	$0$	$0$
$d_{\text{yz}}$	$0$	$0$	$\text{Et2g}$	$0$	$0$
$d_{\text{xz}}$	$0$	$0$	$0$	$\text{Et2g}$	$0$
$d_{\text{xy}}$	$0$	$0$	$0$	$0$	$\text{Et2g}$

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

	$\text{Eeg}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$
	$\text{Eeg}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$
	$\text{Et2g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$
	$\text{Et2g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$
	$\text{Et2g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\ -\frac{3}{4} \sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land (m=-4\lor m=4) \\ -\frac{3}{4} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\ -\frac{39 (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ \frac{39}{280} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_XYZ.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {(-3*Sqrt[5/14]*(2*Ea2u - 3*Et1u + Et2u))/4, k == 4 && (m == -4 || m == 4)}, {(-3*(2*Ea2u - 3*Et1u + Et2u))/4, k == 4 && m == 0}, {(-39*(4*Ea2u + 5*Et1u - 9*Et2u))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}, {(39*(4*Ea2u + 5*Et1u - 9*Et2u))/280, k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Oh_XYZ.Quanty

Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} , 
       {4, 0, (-3/4)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} , 
       {4,-4, (-3/4)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
       {4, 4, (-3/4)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , 
       {6, 0, (39/280)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} , 
       {6,-4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , 
       {6, 4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\frac{1}{8} (5 \text{Et1u}+3 \text{Et2u})$	$0$	$0$	$0$	$\frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u})$	$0$	$0$
${Y_{-2}^{(3)}}$	$0$	$\frac{\text{Ea2u}+\text{Et2u}}{2}$	$0$	$0$	$0$	$\frac{\text{Et2u}-\text{Ea2u}}{2}$	$0$
${Y_{-1}^{(3)}}$	$0$	$0$	$\frac{1}{8} (3 \text{Et1u}+5 \text{Et2u})$	$0$	$0$	$0$	$\frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u})$
${Y_{0}^{(3)}}$	$0$	$0$	$0$	$\text{Et1u}$	$0$	$0$	$0$
${Y_{1}^{(3)}}$	$\frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u})$	$0$	$0$	$0$	$\frac{1}{8} (3 \text{Et1u}+5 \text{Et2u})$	$0$	$0$
${Y_{2}^{(3)}}$	$0$	$\frac{\text{Et2u}-\text{Ea2u}}{2}$	$0$	$0$	$0$	$\frac{\text{Ea2u}+\text{Et2u}}{2}$	$0$
${Y_{3}^{(3)}}$	$0$	$0$	$\frac{1}{8} \sqrt{15} (\text{Et1u}-\text{Et2u})$	$0$	$0$	$0$	$\frac{1}{8} (5 \text{Et1u}+3 \text{Et2u})$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$f_{\text{xyz}}$	$\text{Ea2u}$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{x\left(5x^2-r^2\right)}$	$0$	$\text{Et1u}$	$0$	$0$	$0$	$0$	$0$
$f_{y\left(5y^2-r^2\right)}$	$0$	$0$	$\text{Et1u}$	$0$	$0$	$0$	$0$
$f_{z\left(5z^2-r^2\right)}$	$0$	$0$	$0$	$\text{Et1u}$	$0$	$0$	$0$
$f_{x\left(y^2-z^2\right)}$	$0$	$0$	$0$	$0$	$\text{Et2u}$	$0$	$0$
$f_{y\left(z^2-x^2\right)}$	$0$	$0$	$0$	$0$	$0$	$\text{Et2u}$	$0$
$f_{z\left(x^2-y^2\right)}$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Et2u}$

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

	$\text{Ea2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$
	$\text{Et1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$
	$\text{Et2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$
	$\text{Et2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$
	$\text{Et2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$

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

$A_{k,m} = \begin{cases} 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \frac{3}{4} \sqrt{\frac{15}{2}} \text{Mt1u} & k=4\land (m=-4\lor m=4) \\ \frac{3 \sqrt{21} \text{Mt1u}}{4} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Oh_XYZ.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -4 && m != 0 && m != 4)}, {(3*Sqrt[15/2]*Mt1u)/4, k == 4 && (m == -4 || m == 4)}}, (3*Sqrt[21]*Mt1u)/4]

Input format suitable for Quanty

Akm_Oh_XYZ.Quanty

Akm = {{4, 0, (3/4)*((sqrt(21))*(Mt1u))} , 
       {4,-4, (3/4)*((sqrt(15/2))*(Mt1u))} , 
       {4, 4, (3/4)*((sqrt(15/2))*(Mt1u))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$-\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt1u}$	$0$	$0$	$0$	$-\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt1u}$
${Y_{0}^{(1)}}$	$0$	$0$	$0$	$\text{Mt1u}$	$0$	$0$	$0$
${Y_{1}^{(1)}}$	$-\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt1u}$	$0$	$0$	$0$	$-\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt1u}$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$p_x$	$0$	$\text{Mt1u}$	$0$	$0$	$0$	$0$	$0$
$p_y$	$0$	$0$	$\text{Mt1u}$	$0$	$0$	$0$	$0$
$p_z$	$0$	$0$	$0$	$\text{Mt1u}$	$0$	$0$	$0$

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups	C₁	C_s	C_i
C_n groups	C₂	C₃	C₄	C₅	C₆	C₇	C₈
D_n groups	D₂	D₃	D₄	D₅	D₆	D₇	D₈
C_nv groups	C_2v	C_3v	C_4v	C_5v	C_6v	C_7v	C_8v
C_nh groups	C_2h	C_3h	C_4h	C_5h	C_6h
D_nh groups	D_2h	D_3h	D_4h	D_5h	D_6h	D_7h	D_8h
D_nd groups	D_2d	D_3d	D_4d	D_5d	D_6d	D_7d	D_8d
S_n groups	S₂	S₄	S₆	S₈	S₁₀	S₁₂
Cubic groups	T	T_h	T_d	O	O_h	I	I_h
Linear groups	C_{$\infty$ v}	D_{$\infty$ h}

−Table of Contents

Orientation XYZ

Symmetry Operations

Different Settings

Character Table

Product Table

Sub Groups with compatible settings

Super Groups with compatible settings

Invariant Potential expanded on renormalized spherical Harmonics

Expansion

Input format suitable for Mathematica (Quanty.nb)

Input format suitable for Quanty

One particle coupling on a basis of spherical harmonics

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

One particle coupling on a basis of symmetry adapted functions

Coupling for a single shell

Potential for s orbitals

Potential for p orbitals

Potential for d orbitals

Potential for f orbitals

Coupling between two shells

Potential for p-f orbital mixing

Table of several point groups