−Table of Contents

Orientation xyz

Symmetry Operations

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

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

Different Settings

Point Group Td with orientation xyz

Character Table

	$\text{E} \,{\text{(1)}}$	$C_3 \,{\text{(8)}}$	$C_2 \,{\text{(3)}}$	$S_4 \,{\text{(6)}}$	$\sigma_d \,{\text{(6)}}$
$A_1$	$1$	$1$	$1$	$1$	$1$
$A_2$	$1$	$1$	$1$	$-1$	$-1$
$\text{E}$	$2$	$-1$	$2$	$0$	$0$
$T_1$	$3$	$0$	$-1$	$1$	$-1$
$T_2$	$3$	$0$	$-1$	$-1$	$1$

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

Super Groups with compatible settings

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

Expansion

$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -i B(3,2) & k=3\land m=-2 \\ i B(3,2) & k=3\land m=2 \\ \sqrt{\frac{5}{14}} A(4,0) & k=4\land (m=-4\lor m=4) \\ A(4,0) & k=4\land m=0 \\ -\sqrt{\frac{7}{2}} A(6,0) & k=6\land (m=-4\lor m=4) \\ A(6,0) & k=6\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(-I)*B[3, 2], k == 3 && m == -2}, {I*B[3, 2], k == 3 && m == 2}, {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_Td_xyz.Quanty

Akm = {{0, 0, A(0,0)} , 
       {3,-2, (-I)*(B(3,2))} , 
       {3, 2, (I)*(B(3,2))} , 
       {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.

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(1)}}$	$\color{darkred}{ 0 }$	$\text{App}(0,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0)$
${Y_{0}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)$	$0$	$\color{darkred}{ \frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$0$	$0$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$
${Y_{1}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0)$	$0$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$
${Y_{-2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$\frac{5}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{i \text{Bdf}(3,2)}{\sqrt{21}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{0}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{1}^{(2)}}$	$0$	$\color{darkred}{ -\frac{1}{7} i \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{i \text{Bdf}(3,2)}{\sqrt{21}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2) }$
${Y_{2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{7} i \sqrt{3} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\frac{5}{21} \text{Add}(4,0)$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-3}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$0$	$0$	$0$	$\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0)$	$0$	$0$
${Y_{-2}^{(3)}}$	$\color{darkred}{ -\frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$\frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0)$	$0$
${Y_{-1}^{(3)}}$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{i \text{Bdf}(3,2)}{\sqrt{21}} }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0)$
${Y_{0}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ -\frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2 i \text{Bdf}(3,2)}{3 \sqrt{7}} }$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$
${Y_{1}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{i \text{Bdf}(3,2)}{\sqrt{21}} }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0)$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$
${Y_{2}^{(3)}}$	$\color{darkred}{ \frac{i \text{Bsf}(3,2)}{\sqrt{7}} }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{5}{33} \text{Aff}(4,0)-\frac{70}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$
${Y_{3}^{(3)}}$	$\color{darkred}{ 0 }$	$-\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{1}{11} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{35}{143} \sqrt{\frac{5}{3}} \text{Aff}(6,0)$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

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

	$\text{s}$	$p_x$	$p_y$	$p_z$	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ 0 }$	$\text{App}(0,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$0$	$0$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$0$
$d_{x^2-y^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{yz}}$	$0$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$	$0$	$0$	$0$	$0$	$\text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{\text{xyz}}$	$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)-\frac{4}{11} \text{Aff}(4,0)+\frac{80}{143} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{x\left(5x^2-r^2\right)}$	$\color{darkred}{ 0 }$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$	$0$
$f_{y\left(5y^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2) }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$	$0$
$f_{z\left(5z^2-r^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2) }$	$0$	$0$	$0$	$\text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$0$
$f_{x\left(y^2-z^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0)$	$0$	$0$
$f_{y\left(z^2-x^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0)$	$0$
$f_{z\left(x^2-y^2\right)}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{2}{33} \text{Aff}(4,0)-\frac{60}{143} \text{Aff}(6,0)$

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

$A_{k,m} = \begin{cases} \text{Ea1} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Ea1, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{0, 0, Ea1} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{0}^{(0)}}$
${Y_{0}^{(0)}}$	$\text{Ea1}$

The Hamiltonian on a basis of symmetric functions

	$\text{s}$
$\text{s}$	$\text{Ea1}$

Rotation matrix used

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

Irriducible representations and their onsite energy

	$\text{Ea1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2 \sqrt{\pi }}$

Potential for p orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \text{Et2} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{Et2, k == 0 && m == 0}}, 0]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{0, 0, Et2} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\text{Et2}$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$\text{Et2}$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$\text{Et2}$

The Hamiltonian on a basis of symmetric functions

	$p_x$	$p_y$	$p_z$
$p_x$	$\text{Et2}$	$0$	$0$
$p_y$	$0$	$\text{Et2}$	$0$
$p_z$	$0$	$0$	$\text{Et2}$

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

	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$

Potential for d orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (2 \text{Ee}+3 \text{Et2}) & k=0\land m=0 \\ 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Ee}-\text{Et2}) & k=4\land (m=-4\lor m=4) \\ \frac{21 (\text{Ee}-\text{Et2})}{10} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(2*Ee + 3*Et2)/5, k == 0 && m == 0}, {0, k != 4 || (m != -4 && m != 0 && m != 4)}, {(3*Sqrt[7/10]*(Ee - Et2))/2, k == 4 && (m == -4 || m == 4)}}, (21*(Ee - Et2))/10]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{0, 0, (1/5)*((2)*(Ee) + (3)*(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)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-2}^{(2)}}$	$\frac{\text{Ee}+\text{Et2}}{2}$	$0$	$0$	$0$	$\frac{\text{Ee}-\text{Et2}}{2}$
${Y_{-1}^{(2)}}$	$0$	$\text{Et2}$	$0$	$0$	$0$
${Y_{0}^{(2)}}$	$0$	$0$	$\text{Ee}$	$0$	$0$
${Y_{1}^{(2)}}$	$0$	$0$	$0$	$\text{Et2}$	$0$
${Y_{2}^{(2)}}$	$\frac{\text{Ee}-\text{Et2}}{2}$	$0$	$0$	$0$	$\frac{\text{Ee}+\text{Et2}}{2}$

The Hamiltonian on a basis of symmetric functions

	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$
$d_{x^2-y^2}$	$\text{Ee}$	$0$	$0$	$0$	$0$
$d_{3z^2-r^2}$	$0$	$\text{Ee}$	$0$	$0$	$0$
$d_{\text{yz}}$	$0$	$0$	$\text{Et2}$	$0$	$0$
$d_{\text{xz}}$	$0$	$0$	$0$	$\text{Et2}$	$0$
$d_{\text{xy}}$	$0$	$0$	$0$	$0$	$\text{Et2}$

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

	$\text{Ee}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$
	$\text{Ee}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$

Potential for f orbitals

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea2}+3 (\text{Et1}+\text{Et2})) & k=0\land m=0 \\ 0 & (k\neq 4\land k\neq 6)\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ -\frac{3}{4} \sqrt{\frac{5}{14}} (2 \text{Ea2}+\text{Et1}-3 \text{Et2}) & k=4\land (m=-4\lor m=4) \\ -\frac{3}{4} (2 \text{Ea2}+\text{Et1}-3 \text{Et2}) & k=4\land m=0 \\ -\frac{39 (4 \text{Ea2}-9 \text{Et1}+5 \text{Et2})}{40 \sqrt{14}} & k=6\land (m=-4\lor m=4) \\ \frac{39}{280} (4 \text{Ea2}-9 \text{Et1}+5 \text{Et2}) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea2 + 3*(Et1 + Et2))/7, k == 0 && m == 0}, {0, (k != 4 && k != 6) || (m != -4 && m != 0 && m != 4)}, {(-3*Sqrt[5/14]*(2*Ea2 + Et1 - 3*Et2))/4, k == 4 && (m == -4 || m == 4)}, {(-3*(2*Ea2 + Et1 - 3*Et2))/4, k == 4 && m == 0}, {(-39*(4*Ea2 - 9*Et1 + 5*Et2))/(40*Sqrt[14]), k == 6 && (m == -4 || m == 4)}}, (39*(4*Ea2 - 9*Et1 + 5*Et2))/280]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{0, 0, (1/7)*(Ea2 + (3)*(Et1 + Et2))} , 
       {4, 0, (-3/4)*((2)*(Ea2) + Et1 + (-3)*(Et2))} , 
       {4,-4, (-3/4)*((sqrt(5/14))*((2)*(Ea2) + Et1 + (-3)*(Et2)))} , 
       {4, 4, (-3/4)*((sqrt(5/14))*((2)*(Ea2) + Et1 + (-3)*(Et2)))} , 
       {6, 0, (39/280)*((4)*(Ea2) + (-9)*(Et1) + (5)*(Et2))} , 
       {6,-4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2) + (-9)*(Et1) + (5)*(Et2)))} , 
       {6, 4, (-39/40)*((1/(sqrt(14)))*((4)*(Ea2) + (-9)*(Et1) + (5)*(Et2)))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\frac{1}{8} (3 \text{Et1}+5 \text{Et2})$	$0$	$0$	$0$	$\frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1})$	$0$	$0$
${Y_{-2}^{(3)}}$	$0$	$\frac{\text{Ea2}+\text{Et1}}{2}$	$0$	$0$	$0$	$\frac{\text{Et1}-\text{Ea2}}{2}$	$0$
${Y_{-1}^{(3)}}$	$0$	$0$	$\frac{1}{8} (5 \text{Et1}+3 \text{Et2})$	$0$	$0$	$0$	$\frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1})$
${Y_{0}^{(3)}}$	$0$	$0$	$0$	$\text{Et2}$	$0$	$0$	$0$
${Y_{1}^{(3)}}$	$\frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1})$	$0$	$0$	$0$	$\frac{1}{8} (5 \text{Et1}+3 \text{Et2})$	$0$	$0$
${Y_{2}^{(3)}}$	$0$	$\frac{\text{Et1}-\text{Ea2}}{2}$	$0$	$0$	$0$	$\frac{\text{Ea2}+\text{Et1}}{2}$	$0$
${Y_{3}^{(3)}}$	$0$	$0$	$\frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1})$	$0$	$0$	$0$	$\frac{1}{8} (3 \text{Et1}+5 \text{Et2})$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$f_{\text{xyz}}$	$\text{Ea2}$	$0$	$0$	$0$	$0$	$0$	$0$
$f_{x\left(5x^2-r^2\right)}$	$0$	$\text{Et2}$	$0$	$0$	$0$	$0$	$0$
$f_{y\left(5y^2-r^2\right)}$	$0$	$0$	$\text{Et2}$	$0$	$0$	$0$	$0$
$f_{z\left(5z^2-r^2\right)}$	$0$	$0$	$0$	$\text{Et2}$	$0$	$0$	$0$
$f_{x\left(y^2-z^2\right)}$	$0$	$0$	$0$	$0$	$\text{Et1}$	$0$	$0$
$f_{y\left(z^2-x^2\right)}$	$0$	$0$	$0$	$0$	$0$	$\text{Et1}$	$0$
$f_{z\left(x^2-y^2\right)}$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Et1}$

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

	$\text{Ea2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$
	$\text{Et2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$
	$\text{Et1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$
	$\text{Et1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$
	$\text{Et1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$

Coupling between two shells

Click on one of the subsections to expand it or

Potential for s-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ i \sqrt{\frac{7}{2}} \text{Ma1} & k=3\land m=-2 \\ -i \sqrt{\frac{7}{2}} \text{Ma1} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 2)}, {I*Sqrt[7/2]*Ma1, k == 3 && m == -2}}, (-I)*Sqrt[7/2]*Ma1]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{3, 2, (-I)*((sqrt(7/2))*(Ma1))} , 
       {3,-2, (I)*((sqrt(7/2))*(Ma1))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$0$	$-\frac{i \text{Ma1}}{\sqrt{2}}$	$0$	$0$	$0$	$\frac{i \text{Ma1}}{\sqrt{2}}$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$\text{s}$	$\text{Ma1}$	$0$	$0$	$0$	$0$	$0$	$0$

Potential for p-d orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ \frac{7 i \text{Mt2}}{\sqrt{6}} & k=3\land m=-2 \\ -\frac{7 i \text{Mt2}}{\sqrt{6}} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 2)}, {((7*I)*Mt2)/Sqrt[6], k == 3 && m == -2}}, ((-7*I)*Mt2)/Sqrt[6]]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{3, 2, (-7*I)*((1/(sqrt(6)))*(Mt2))} , 
       {3,-2, (7*I)*((1/(sqrt(6)))*(Mt2))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$0$	$-i \text{Mt2}$	$0$
${Y_{0}^{(1)}}$	$-\frac{i \text{Mt2}}{\sqrt{2}}$	$0$	$0$	$0$	$\frac{i \text{Mt2}}{\sqrt{2}}$
${Y_{1}^{(1)}}$	$0$	$i \text{Mt2}$	$0$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$d_{x^2-y^2}$	$d_{3z^2-r^2}$	$d_{\text{yz}}$	$d_{\text{xz}}$	$d_{\text{xy}}$
$p_x$	$0$	$0$	$\text{Mt2}$	$0$	$0$
$p_y$	$0$	$0$	$0$	$\text{Mt2}$	$0$
$p_z$	$0$	$0$	$0$	$0$	$\text{Mt2}$

Potential for p-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \frac{3}{4} \sqrt{\frac{15}{2}} \text{Mt2} & k=4\land (m=-4\lor m=4) \\ \frac{3 \sqrt{21} \text{Mt2}}{4} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -4 && m != 0 && m != 4)}, {(3*Sqrt[15/2]*Mt2)/4, k == 4 && (m == -4 || m == 4)}}, (3*Sqrt[21]*Mt2)/4]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{4, 0, (3/4)*((sqrt(21))*(Mt2))} , 
       {4,-4, (3/4)*((sqrt(15/2))*(Mt2))} , 
       {4, 4, (3/4)*((sqrt(15/2))*(Mt2))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$-\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt2}$	$0$	$0$	$0$	$-\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt2}$
${Y_{0}^{(1)}}$	$0$	$0$	$0$	$\text{Mt2}$	$0$	$0$	$0$
${Y_{1}^{(1)}}$	$-\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt2}$	$0$	$0$	$0$	$-\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt2}$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$p_x$	$0$	$\text{Mt2}$	$0$	$0$	$0$	$0$	$0$
$p_y$	$0$	$0$	$\text{Mt2}$	$0$	$0$	$0$	$0$
$p_z$	$0$	$0$	$0$	$\text{Mt2}$	$0$	$0$	$0$

Potential for d-f orbital mixing

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ -\frac{3}{2} i \sqrt{\frac{7}{2}} \text{Mt2} & k=3\land m=-2 \\ \frac{3}{2} i \sqrt{\frac{7}{2}} \text{Mt2} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_Td_xyz.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 2)}, {((-3*I)/2)*Sqrt[7/2]*Mt2, k == 3 && m == -2}}, ((3*I)/2)*Sqrt[7/2]*Mt2]

Input format suitable for Quanty

Akm_Td_xyz.Quanty

Akm = {{3,-2, (-3/2*I)*((sqrt(7/2))*(Mt2))} , 
       {3, 2, (3/2*I)*((sqrt(7/2))*(Mt2))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-2}^{(2)}}$	$0$	$0$	$0$	$\frac{i \text{Mt2}}{\sqrt{2}}$	$0$	$0$	$0$
${Y_{-1}^{(2)}}$	$\frac{1}{2} i \sqrt{\frac{5}{2}} \text{Mt2}$	$0$	$0$	$0$	$\frac{1}{2} i \sqrt{\frac{3}{2}} \text{Mt2}$	$0$	$0$
${Y_{0}^{(2)}}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
${Y_{1}^{(2)}}$	$0$	$0$	$-\frac{1}{2} i \sqrt{\frac{3}{2}} \text{Mt2}$	$0$	$0$	$0$	$-\frac{1}{2} i \sqrt{\frac{5}{2}} \text{Mt2}$
${Y_{2}^{(2)}}$	$0$	$0$	$0$	$-\frac{i \text{Mt2}}{\sqrt{2}}$	$0$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{\text{xyz}}$	$f_{x\left(5x^2-r^2\right)}$	$f_{y\left(5y^2-r^2\right)}$	$f_{z\left(5z^2-r^2\right)}$	$f_{x\left(y^2-z^2\right)}$	$f_{y\left(z^2-x^2\right)}$	$f_{z\left(x^2-y^2\right)}$
$d_{x^2-y^2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{3z^2-r^2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{\text{yz}}$	$0$	$\text{Mt2}$	$0$	$0$	$0$	$0$	$0$
$d_{\text{xz}}$	$0$	$0$	$\text{Mt2}$	$0$	$0$	$0$	$0$
$d_{\text{xy}}$	$0$	$0$	$0$	$\text{Mt2}$	$0$	$0$	$0$

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}