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Orientation Zx_A

The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry eg $\sigma$ and the eg orbitals that descend from the t2g irreducible representation eg $\pi$ orbitals. (The mixing is given by the parameter Meg.) ###

As one can see in the list of supergroups of D3d, there are two different orientations of Oh that are a supergroup of this orientation of D3d. The different orientations of Oh with respect to D3d do however change the definitions of the eg $\pi$ and eg $\sigma$ orbitals. We include three different representations of the orbitals and potentials for each setting of D3d symmetry. The orientation without additional letter takes the tesseral harmonics as a basis. This basis does not relate to the states in Oh symmetry. The orientation with an additional A or B relate to the two different supergroup representations of the Oh point group.

The parameterisation A of the orientation Zx is related to the orientation Sqrt[2]1z of the Oh pointgroup.

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

In the D3d Point Group, with orientation Zx_A there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$\sigma _d$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,

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

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

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

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

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

===== 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 D3d Point group with orientation Zx_A the form of the expansion coefficients is:

==== Expansion ====

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

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

Akm_D3d_Zx_A.Quanty.nb

Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {I*B[4, 3], k == 4 && (m == -3 || m == 3)}, {A[4, 0], k == 4 && m == 0}, {A[6, 6], k == 6 && (m == -6 || m == 6)}, {I*B[6, 3], k == 6 && (m == -3 || m == 3)}, {A[6, 0], k == 6 && m == 0}}, 0]

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

Akm_D3d_Zx_A.Quanty

Akm = {{0, 0, A(0,0)} , 
       {2, 0, A(2,0)} , 
       {4, 0, A(4,0)} , 
       {4,-3, (I)*(B(4,3))} , 
       {4, 3, (I)*(B(4,3))} , 
       {6, 0, A(6,0)} , 
       {6,-3, (I)*(B(6,3))} , 
       {6, 3, (I)*(B(6,3))} , 
       {6,-6, A(6,6)} , 
       {6, 6, A(6,6)} }

==== 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.

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{0}^{(0)}}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(1)}}$	$\color{darkred}{ 0 }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\frac{1}{3} i \text{Bpf}(4,3)$	$0$
${Y_{0}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$0$	$-\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$
${Y_{1}^{(1)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{1}{3} i \text{Bpf}(4,3)$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$
${Y_{-2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$0$	$0$	$-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$0$	$\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{0}^{(2)}}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{1}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$0$	$\text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{2}^{(2)}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{1}{3} i \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$0$	$\text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
${Y_{-3}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$	$0$	$0$	$\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)-\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)$	$0$	$0$	$-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$
${Y_{-2}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{3} i \text{Bpf}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$	$0$	$-\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)-\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$0$	$0$
${Y_{-1}^{(3)}}$	$\color{darkred}{ 0 }$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$	$\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$0$
${Y_{0}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)-\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)$	$0$	$0$	$\text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0)$	$0$	$0$	$\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)-\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)$
${Y_{1}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0)$	$0$	$0$
${Y_{2}^{(3)}}$	$\color{darkred}{ 0 }$	$-\frac{1}{3} i \text{Bpf}(4,3)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{33} i \sqrt{14} \text{Bff}(4,3)-\frac{5}{143} i \sqrt{42} \text{Bff}(6,3)$	$0$	$0$	$\text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0)$	$0$
${Y_{3}^{(3)}}$	$\color{darkred}{ 0 }$	$0$	$\frac{i \text{Bpf}(4,3)}{3 \sqrt{3}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$	$0$	$0$	$\frac{10}{143} i \sqrt{\frac{7}{3}} \text{Bff}(6,3)-\frac{1}{11} i \sqrt{7} \text{Bff}(4,3)$	$0$	$0$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)$

==== 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

	${Y_{0}^{(0)}}$	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$\text{s}$	$1$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{2}}$	$0$	$-\frac{1}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$\frac{i}{\sqrt{2}}$	$0$	$\frac{i}{\sqrt{2}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$1$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$d_{\sqrt{2}\text{xz}-\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i}{\sqrt{6}}$	$\frac{1}{\sqrt{3}}$	$0$	$-\frac{1}{\sqrt{3}}$	$\frac{i}{\sqrt{6}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{1}{\sqrt{6}}$	$\frac{i}{\sqrt{3}}$	$0$	$\frac{i}{\sqrt{3}}$	$-\frac{1}{\sqrt{6}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}-\sqrt{2}\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i}{\sqrt{3}}$	$-\frac{1}{\sqrt{6}}$	$0$	$\frac{1}{\sqrt{6}}$	$\frac{i}{\sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{1}{\sqrt{3}}$	$-\frac{i}{\sqrt{6}}$	$0$	$-\frac{i}{\sqrt{6}}$	$-\frac{1}{\sqrt{3}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$1$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{i \sqrt{2}}{3}$	$0$	$0$	$\frac{\sqrt{5}}{3}$	$0$	$0$	$-\frac{i \sqrt{2}}{3}$
$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{1}{2} i \sqrt{\frac{5}{3}}$	$-\frac{1}{2 \sqrt{3}}$	$0$	$\frac{1}{2 \sqrt{3}}$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$0$
$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$-\frac{i}{2 \sqrt{3}}$	$0$	$-\frac{i}{2 \sqrt{3}}$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$0$
$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{3} i \sqrt{\frac{5}{2}}$	$0$	$0$	$\frac{2}{3}$	$0$	$0$	$\frac{1}{3} i \sqrt{\frac{5}{2}}$
$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{i}{2 \sqrt{3}}$	$-\frac{\sqrt{\frac{5}{3}}}{2}$	$0$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$\frac{i}{2 \sqrt{3}}$	$0$
$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{1}{2 \sqrt{3}}$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$0$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$-\frac{1}{2 \sqrt{3}}$	$0$
$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$0$	$0$	$-\frac{1}{\sqrt{2}}$

==== One particle coupling on a basis of symmetry adapted functions ====

After rotation we find

	$\text{s}$	$p_x$	$p_y$	$p_z$	$d_{\sqrt{2}\text{xz}-\text{xy}}$	$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$d_{\text{xz}-\sqrt{2}\text{xy}}$	$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$d_{3z^2-r^2}$	$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$
$\text{s}$	$\text{Ass}(0,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$p_x$	$\color{darkred}{ 0 }$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,3)$	$0$	$0$	$-\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Bpf}(4,3)}{3 \sqrt{6}}$	$0$	$0$
$p_y$	$\color{darkred}{ 0 }$	$0$	$\text{App}(0,0)-\frac{1}{5} \text{App}(2,0)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,3)$	$0$	$0$	$-\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Bpf}(4,3)}{3 \sqrt{6}}$	$0$
$p_z$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{App}(0,0)+\frac{2}{5} \text{App}(2,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{4}{9} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{2}{9} \sqrt{\frac{2}{3}} \text{Bpf}(4,3)$	$0$	$0$	$\frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{8 \text{Apf}(4,0)}{9 \sqrt{21}}+\frac{1}{9} \sqrt{\frac{10}{3}} \text{Bpf}(4,3)$	$0$	$0$	$0$
$d_{\sqrt{2}\text{xz}-\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Add}(0,0)-\frac{1}{9} \text{Add}(4,0)-\frac{2}{9} \sqrt{\frac{10}{7}} \text{Bdd}(4,3)$	$0$	$-\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Add}(0,0)-\frac{1}{9} \text{Add}(4,0)-\frac{2}{9} \sqrt{\frac{10}{7}} \text{Bdd}(4,3)$	$0$	$-\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{\text{xz}-\sqrt{2}\text{xy}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$-\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)-\frac{2}{63} \text{Add}(4,0)+\frac{2}{9} \sqrt{\frac{10}{7}} \text{Bdd}(4,3)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$-\frac{1}{7} \sqrt{2} \text{Add}(2,0)+\frac{5}{63} \sqrt{2} \text{Add}(4,0)-\frac{1}{9} \sqrt{\frac{5}{7}} \text{Bdd}(4,3)$	$0$	$\text{Add}(0,0)-\frac{1}{7} \text{Add}(2,0)-\frac{2}{63} \text{Add}(4,0)+\frac{2}{9} \sqrt{\frac{10}{7}} \text{Bdd}(4,3)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$d_{3z^2-r^2}$	$\frac{\text{Asd}(2,0)}{\sqrt{5}}$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$\text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$
$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{4}{9} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{2}{9} \sqrt{\frac{2}{3}} \text{Bpf}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\text{Aff}(0,0)+\frac{14}{99} \text{Aff}(4,0)+\frac{160 \text{Aff}(6,0)}{1287}-\frac{40}{117} \sqrt{\frac{7}{33}} \text{Aff}(6,6)+\frac{4}{99} \sqrt{70} \text{Bff}(4,3)-\frac{40 \sqrt{\frac{70}{3}} \text{Bff}(6,3)}{1287}$	$0$	$0$	$\frac{2 \text{Aff}(2,0)}{3 \sqrt{5}}+\frac{2}{99} \sqrt{5} \text{Aff}(4,0)+\frac{70 \sqrt{5} \text{Aff}(6,0)}{1287}+\frac{20}{117} \sqrt{\frac{35}{33}} \text{Aff}(6,6)-\frac{1}{99} \sqrt{14} \text{Bff}(4,3)+\frac{10 \sqrt{\frac{14}{3}} \text{Bff}(6,3)}{1287}$	$0$	$0$	$0$
$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$\color{darkred}{ 0 }$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,3)$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\text{Aff}(0,0)+\frac{1}{30} \text{Aff}(2,0)-\frac{17}{99} \text{Aff}(4,0)+\frac{25}{858} \text{Aff}(6,0)-\frac{1}{99} \sqrt{70} \text{Bff}(4,3)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,3)$	$0$	$0$	$\frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{2}{99} \sqrt{14} \text{Bff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)$	$0$	$0$
$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$\color{darkred}{ 0 }$	$0$	$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{3 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,3)$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{30} \text{Aff}(2,0)-\frac{17}{99} \text{Aff}(4,0)+\frac{25}{858} \text{Aff}(6,0)-\frac{1}{99} \sqrt{70} \text{Bff}(4,3)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,3)$	$0$	$0$	$\frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{2}{99} \sqrt{14} \text{Bff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)$	$0$
$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{8 \text{Apf}(4,0)}{9 \sqrt{21}}+\frac{1}{9} \sqrt{\frac{10}{3}} \text{Bpf}(4,3)$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\frac{2 \text{Aff}(2,0)}{3 \sqrt{5}}+\frac{2}{99} \sqrt{5} \text{Aff}(4,0)+\frac{70 \sqrt{5} \text{Aff}(6,0)}{1287}+\frac{20}{117} \sqrt{\frac{35}{33}} \text{Aff}(6,6)-\frac{1}{99} \sqrt{14} \text{Bff}(4,3)+\frac{10 \sqrt{\frac{14}{3}} \text{Bff}(6,3)}{1287}$	$0$	$0$	$\text{Aff}(0,0)-\frac{1}{15} \text{Aff}(2,0)+\frac{13}{99} \text{Aff}(4,0)+\frac{125 \text{Aff}(6,0)}{1287}-\frac{50}{117} \sqrt{\frac{7}{33}} \text{Aff}(6,6)-\frac{4}{99} \sqrt{70} \text{Bff}(4,3)+\frac{40 \sqrt{\frac{70}{3}} \text{Bff}(6,3)}{1287}$	$0$	$0$	$0$
$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$-\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Bpf}(4,3)}{3 \sqrt{6}}$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$\frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{2}{99} \sqrt{14} \text{Bff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{6} \text{Aff}(2,0)-\frac{1}{99} \text{Aff}(4,0)-\frac{115}{858} \text{Aff}(6,0)+\frac{1}{99} \sqrt{70} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,3)$	$0$	$0$
$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$\color{darkred}{ 0 }$	$0$	$-\sqrt{\frac{3}{35}} \text{Apf}(2,0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,0)-\frac{\text{Bpf}(4,3)}{3 \sqrt{6}}$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$\frac{\text{Aff}(2,0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,0)-\frac{35}{858} \sqrt{5} \text{Aff}(6,0)-\frac{2}{99} \sqrt{14} \text{Bff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)$	$0$	$0$	$\text{Aff}(0,0)+\frac{1}{6} \text{Aff}(2,0)-\frac{1}{99} \text{Aff}(4,0)-\frac{115}{858} \text{Aff}(6,0)+\frac{1}{99} \sqrt{70} \text{Bff}(4,3)+\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,3)$	$0$
$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$\color{darkred}{ 0 }$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)+\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6)$

===== 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_D3d_Zx_A.Quanty.nb

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

Input format suitable for Quanty

Akm_D3d_Zx_A.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

$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\ \frac{5 (\text{Ea2u}-\text{Eeu})}{3} & k=2\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Zx_A.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(5*(Ea2u - Eeu))/3, k == 2 && m == 0}}, 0]

Input format suitable for Quanty

Akm_D3d_Zx_A.Quanty

Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , 
       {2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-1}^{(1)}}$	${Y_{0}^{(1)}}$	${Y_{1}^{(1)}}$
${Y_{-1}^{(1)}}$	$\text{Eeu}$	$0$	$0$
${Y_{0}^{(1)}}$	$0$	$\text{Ea2u}$	$0$
${Y_{1}^{(1)}}$	$0$	$0$	$\text{Eeu}$

The Hamiltonian on a basis of symmetric functions

	$p_x$	$p_y$	$p_z$
$p_x$	$\text{Eeu}$	$0$	$0$
$p_y$	$0$	$\text{Eeu}$	$0$
$p_z$	$0$	$0$	$\text{Ea2u}$

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

	$\text{Eeu}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$
	$\text{Eeu}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$
	$\text{Ea2u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$

==== Potential for d orbitals ====

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea1g}+2 (\text{Eeg$\pi $}+\text{Eeg$\sigma $})) & k=0\land m=0 \text{Ea1g}-\text{Eeg$\pi $}-2 \sqrt{2} \text{Meg} & k=2\land m=0 -i \sqrt{\frac{7}{10}} \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right) & k=4\land (m=-3\lor m=3) \frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg$\pi $}-7 \text{Eeg$\sigma $}+10 \sqrt{2} \text{Meg}\right) & k=4\land m=0 \end{cases}$ ### </hidden> <hidden **Input format suitable for Mathematica (Quanty.nb)** > ### <code Quanty Akm_D3d_Zx_A.Quanty.nb> Akm[k_,m_]:=Piecewise[{{(Ea1g + 2*(Eeg\[Pi] + Eeg\[Sigma]))/5, k == 0 && m == 0}, {Ea1g - Eeg\[Pi] - 2*Sqrt[2]*Meg, k == 2 && m == 0}, {(-I)*Sqrt[7/10]*(-2*Eeg\[Pi] + 2*Eeg\[Sigma] + Sqrt[2]*Meg), k == 4 && (m == -3 || m == 3)}, {(9*Ea1g - 2*Eeg\[Pi] - 7*Eeg\[Sigma] + 10*Sqrt[2]*Meg)/5, k == 4 && m == 0}}, 0] </code> ### </hidden><hidden **Input format suitable for Quanty** > ### <code Quanty Akm_D3d_Zx_A.Quanty> Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg\[Pi] + Eeg\[Sigma]))} , {2, 0, Ea1g + (-1)*(Eeg\[Pi]) + (-2)*((sqrt(2))*(Meg))} , {4, 0, (1/5)*((9)*(Ea1g) + (-2)*(Eeg\[Pi]) + (-7)*(Eeg\[Sigma]) + (10)*((sqrt(2))*(Meg)))} , {4,-3, (-I)*((sqrt(7/10))*((-2)*(Eeg\[Pi]) + (2)*(Eeg\[Sigma]) + (sqrt(2))*(Meg)))} , {4, 3, (-I)*((sqrt(7/10))*((-2)*(Eeg\[Pi]) + (2)*(Eeg\[Sigma]) + (sqrt(2))*(Meg)))} } </code> ### </hidden> <hidden **The Hamiltonian on a basis of spherical Harmonics** > ### | ^ ${Y_{-2}^{(2)}}$ ^ ${Y_{-1}^{(2)}}$ ^ ${Y_{0}^{(2)}}$ ^ ${Y_{1}^{(2)}}$ ^ ${Y_{2}^{(2)}}$ ^

${Y_{-2}^{(2)}}$	$\frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right)$	$0$	$0$	$\frac{i \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right)}{3 \sqrt{2}}$	$0$
${Y_{-1}^{(2)}}$	$0$	$\frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right)$	$0$	$0$	$\frac{1}{3} i \left(\sqrt{2} \text{Eeg$\pi $}-\sqrt{2} \text{Eeg$\sigma $}-\text{Meg}\right)$
${Y_{0}^{(2)}}$	$0$	$0$	$\text{Ea1g}$	$0$	$0$
${Y_{1}^{(2)}}$	$\frac{1}{3} i \left(\sqrt{2} \text{Eeg$\pi $}-\sqrt{2} \text{Eeg$\sigma $}-\text{Meg}\right)$	$0$	$0$	$\frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right)$	$0$
${Y_{2}^{(2)}}$	$0$	$\frac{i \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right)}{3 \sqrt{2}}$	$0$	$0$	$\frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right)$

The Hamiltonian on a basis of symmetric functions

	$d_{\sqrt{2}\text{xz}-\text{xy}}$	$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$d_{\text{xz}-\sqrt{2}\text{xy}}$	$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$d_{3z^2-r^2}$
$d_{\sqrt{2}\text{xz}-\text{xy}}$	$\text{Eeg$\sigma $}$	$0$	$\text{Meg}$	$0$	$0$
$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$0$	$\text{Eeg$\sigma $}$	$0$	$\text{Meg}$	$0$
$d_{\text{xz}-\sqrt{2}\text{xy}}$	$\text{Meg}$	$0$	$\text{Eeg$\pi $}$	$0$	$0$
$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$0$	$\text{Meg}$	$0$	$\text{Eeg$\pi $}$	$0$
$d_{3z^2-r^2}$	$0$	$0$	$0$	$0$	$\text{Ea1g}$

Rotation matrix used

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
$d_{\sqrt{2}\text{xz}-\text{xy}}$	$-\frac{i}{\sqrt{6}}$	$\frac{1}{\sqrt{3}}$	$0$	$-\frac{1}{\sqrt{3}}$	$\frac{i}{\sqrt{6}}$
$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$-\frac{1}{\sqrt{6}}$	$\frac{i}{\sqrt{3}}$	$0$	$\frac{i}{\sqrt{3}}$	$-\frac{1}{\sqrt{6}}$
$d_{\text{xz}-\sqrt{2}\text{xy}}$	$-\frac{i}{\sqrt{3}}$	$-\frac{1}{\sqrt{6}}$	$0$	$\frac{1}{\sqrt{6}}$	$\frac{i}{\sqrt{3}}$
$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$-\frac{1}{\sqrt{3}}$	$-\frac{i}{\sqrt{6}}$	$0$	$-\frac{i}{\sqrt{6}}$	$-\frac{1}{\sqrt{3}}$
$d_{3z^2-r^2}$	$0$	$0$	$1$	$0$	$0$

Irriducible representations and their onsite energy

	$\text{Eeg$\sigma $}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sqrt{2} \cos (\theta )-\sin (\theta ) \sin (\phi )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(y-\sqrt{2} z\right)$
	$\text{Eeg$\sigma $}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sin (\theta ) \cos (2 \phi )-2 \sqrt{2} \cos (\theta ) \sin (\phi )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(y \left(y+2 \sqrt{2} z\right)-x^2\right)$
	$\text{Eeg$\pi $}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sqrt{2} \sin (\theta ) \sin (\phi )+\cos (\theta )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(\sqrt{2} y+z\right)$
	$\text{Eeg$\pi $}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (\theta ) \cos (2 \phi )+2 \cos (\theta ) \sin (\phi )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(\sqrt{2} x^2+y \left(2 z-\sqrt{2} y\right)\right)$
	$\text{Ea1g}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$

==== Potential for f orbitals ====

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\ -\frac{5}{28} \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}-4 \sqrt{5} \text{Ma2u}-2 \sqrt{5} \text{Meu}\right) & k=2\land m=0 \\ \frac{i \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}-\text{Ma2u}-4 \text{Meu}\right)}{\sqrt{14}} & k=4\land (m=-3\lor m=3) \\ \frac{1}{14} \left(9 \text{Ea1u}+14 \text{Ea2u1}+13 \text{Ea2u2}-34 \text{Eeu1}-2 \text{Eeu2}+4 \sqrt{5} \text{Ma2u}+16 \sqrt{5} \text{Meu}\right) & k=4\land m=0 \\ \frac{13}{60} \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) & k=6\land (m=-6\lor m=6) \\ -\frac{13 i \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}-2 \text{Ma2u}+36 \text{Meu}\right)}{30 \sqrt{42}} & k=6\land (m=-3\lor m=3) \\ -\frac{13}{420} \left(3 \text{Ea1u}-32 \text{Ea2u1}-25 \text{Ea2u2}-15 \text{Eeu1}+69 \text{Eeu2}-28 \sqrt{5} \text{Ma2u}+42 \sqrt{5} \text{Meu}\right) & k=6\land m=0 \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Zx_A.Quanty.nb

Akm[k_,m_]:=Piecewise[{{(Ea1u + Ea2u1 + Ea2u2 + 2*Eeu1 + 2*Eeu2)/7, k == 0 && m == 0}, {(-5*(5*Ea1u + Ea2u2 - Eeu1 - 5*Eeu2 - 4*Sqrt[5]*Ma2u - 2*Sqrt[5]*Meu))/28, k == 2 && m == 0}, {(I*(2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 - Ma2u - 4*Meu))/Sqrt[14], k == 4 && (m == -3 || m == 3)}, {(9*Ea1u + 14*Ea2u1 + 13*Ea2u2 - 34*Eeu1 - 2*Eeu2 + 4*Sqrt[5]*Ma2u + 16*Sqrt[5]*Meu)/14, k == 4 && m == 0}, {(13*Sqrt[11/21]*(9*Ea1u - 4*Ea2u1 - 5*Ea2u2 + 4*Sqrt[5]*Ma2u))/60, k == 6 && (m == -6 || m == 6)}, {(((-13*I)/30)*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 - 2*Ma2u + 36*Meu))/Sqrt[42], k == 6 && (m == -3 || m == 3)}, {(-13*(3*Ea1u - 32*Ea2u1 - 25*Ea2u2 - 15*Eeu1 + 69*Eeu2 - 28*Sqrt[5]*Ma2u + 42*Sqrt[5]*Meu))/420, k == 6 && m == 0}}, 0]

Input format suitable for Quanty

Akm_D3d_Zx_A.Quanty

Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1) + (2)*(Eeu2))} , 
       {2, 0, (-5/28)*((5)*(Ea1u) + Ea2u2 + (-1)*(Eeu1) + (-5)*(Eeu2) + (-4)*((sqrt(5))*(Ma2u)) + (-2)*((sqrt(5))*(Meu)))} , 
       {4, 0, (1/14)*((9)*(Ea1u) + (14)*(Ea2u1) + (13)*(Ea2u2) + (-34)*(Eeu1) + (-2)*(Eeu2) + (4)*((sqrt(5))*(Ma2u)) + (16)*((sqrt(5))*(Meu)))} , 
       {4,-3, (I)*((1/(sqrt(14)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + (-1)*(Ma2u) + (-4)*(Meu)))} , 
       {4, 3, (I)*((1/(sqrt(14)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + (-1)*(Ma2u) + (-4)*(Meu)))} , 
       {6, 0, (-13/420)*((3)*(Ea1u) + (-32)*(Ea2u1) + (-25)*(Ea2u2) + (-15)*(Eeu1) + (69)*(Eeu2) + (-28)*((sqrt(5))*(Ma2u)) + (42)*((sqrt(5))*(Meu)))} , 
       {6,-3, (-13/30*I)*((1/(sqrt(42)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (-2)*(Ma2u) + (36)*(Meu)))} , 
       {6, 3, (-13/30*I)*((1/(sqrt(42)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (-2)*(Ma2u) + (36)*(Meu)))} , 
       {6,-6, (13/60)*((sqrt(11/21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (4)*((sqrt(5))*(Ma2u))))} , 
       {6, 6, (13/60)*((sqrt(11/21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (4)*((sqrt(5))*(Ma2u))))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-3}^{(3)}}$	$\frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right)$	$0$	$0$	$-\frac{i \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right)}{9 \sqrt{2}}$	$0$	$0$	$\frac{1}{18} \left(-9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right)$
${Y_{-2}^{(3)}}$	$0$	$\frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}-2 \sqrt{5} \text{Meu}\right)$	$0$	$0$	$\frac{1}{6} i \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$	$0$	$0$
${Y_{-1}^{(3)}}$	$0$	$0$	$\frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}+2 \sqrt{5} \text{Meu}\right)$	$0$	$0$	$-\frac{1}{6} i \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$	$0$
${Y_{0}^{(3)}}$	$\frac{i \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right)}{9 \sqrt{2}}$	$0$	$0$	$\frac{1}{9} \left(5 \text{Ea2u1}+4 \left(\text{Ea2u2}+\sqrt{5} \text{Ma2u}\right)\right)$	$0$	$0$	$\frac{i \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right)}{9 \sqrt{2}}$
${Y_{1}^{(3)}}$	$0$	$-\frac{1}{6} i \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$	$0$	$0$	$\frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}+2 \sqrt{5} \text{Meu}\right)$	$0$	$0$
${Y_{2}^{(3)}}$	$0$	$0$	$\frac{1}{6} i \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$	$0$	$0$	$\frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}-2 \sqrt{5} \text{Meu}\right)$	$0$
${Y_{3}^{(3)}}$	$\frac{1}{18} \left(-9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right)$	$0$	$0$	$-\frac{i \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\text{Ma2u}\right)}{9 \sqrt{2}}$	$0$	$0$	$\frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right)$

The Hamiltonian on a basis of symmetric functions

	$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$
$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$\text{Ea2u1}$	$0$	$0$	$\text{Ma2u}$	$0$	$0$	$0$
$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$0$	$\text{Eeu1}$	$0$	$0$	$\text{Meu}$	$0$	$0$
$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$0$	$0$	$\text{Eeu1}$	$0$	$0$	$\text{Meu}$	$0$
$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$\text{Ma2u}$	$0$	$0$	$\text{Ea2u2}$	$0$	$0$	$0$
$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$0$	$\text{Meu}$	$0$	$0$	$\text{Eeu2}$	$0$	$0$
$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$0$	$0$	$\text{Meu}$	$0$	$0$	$\text{Eeu2}$	$0$
$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$	$0$	$0$	$0$	$0$	$0$	$0$	$\text{Ea1u}$

Rotation matrix used

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$-\frac{i \sqrt{2}}{3}$	$0$	$0$	$\frac{\sqrt{5}}{3}$	$0$	$0$	$-\frac{i \sqrt{2}}{3}$
$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$0$	$\frac{1}{2} i \sqrt{\frac{5}{3}}$	$-\frac{1}{2 \sqrt{3}}$	$0$	$\frac{1}{2 \sqrt{3}}$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$0$
$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$0$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$-\frac{i}{2 \sqrt{3}}$	$0$	$-\frac{i}{2 \sqrt{3}}$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$0$
$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$\frac{1}{3} i \sqrt{\frac{5}{2}}$	$0$	$0$	$\frac{2}{3}$	$0$	$0$	$\frac{1}{3} i \sqrt{\frac{5}{2}}$
$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$0$	$-\frac{i}{2 \sqrt{3}}$	$-\frac{\sqrt{\frac{5}{3}}}{2}$	$0$	$\frac{\sqrt{\frac{5}{3}}}{2}$	$\frac{i}{2 \sqrt{3}}$	$0$
$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$0$	$-\frac{1}{2 \sqrt{3}}$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$0$	$-\frac{1}{2} i \sqrt{\frac{5}{3}}$	$-\frac{1}{2 \sqrt{3}}$	$0$
$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$	$\frac{1}{\sqrt{2}}$	$0$	$0$	$0$	$0$	$0$	$-\frac{1}{\sqrt{2}}$

Irriducible representations and their onsite energy

	$\text{Ea2u1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{24} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(i \sqrt{2} \left(-1+e^{6 i \phi }\right) \sin ^3(\theta )+e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(-3 \sqrt{2} x^2 y+\sqrt{2} y^3+5 z^3-3 z\right)$
	$\text{Eeu1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(-10 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{7}{\pi }} x \left(10 \sqrt{2} y z-5 z^2+1\right)$
	$\text{Eeu1}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \left(5 \sqrt{2} \sin (2 \theta ) \cos (2 \phi )-(5 \cos (2 \theta )+3) \sin (\phi )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 \sqrt{2} x^2 z-5 \sqrt{2} y^2 z-5 y z^2+y\right)$
	$\text{Ea2u2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 \left(\sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+\cos (3 \theta )\right)+3 \cos (\theta )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(15 \sqrt{2} x^2 y-5 \sqrt{2} y^3+4 z \left(5 z^2-3\right)\right)$
	$\text{Eeu2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{8} \sqrt{\frac{35}{\pi }} x \left(2 \sqrt{2} y z+5 z^2-1\right)$
	$\text{Eeu2}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (2 \theta ) \cos (2 \phi )+(5 \cos (2 \theta )+3) \sin (\phi )\right)$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(-\sqrt{2} x^2 z+\sqrt{2} y^2 z-5 y z^2+y\right)$
	$\text{Ea1u}$
$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$
$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$	$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$

===== Coupling between two shells =====

Click on one of the subsections to expand it or

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

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & k\neq 2\lor m\neq 0 \\ \sqrt{5} \text{Ma1g} & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Zx_A.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, Sqrt[5]*Ma1g]

Input format suitable for Quanty

Akm_D3d_Zx_A.Quanty

Akm = {{2, 0, (sqrt(5))*(Ma1g)} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-2}^{(2)}}$	${Y_{-1}^{(2)}}$	${Y_{0}^{(2)}}$	${Y_{1}^{(2)}}$	${Y_{2}^{(2)}}$
${Y_{0}^{(0)}}$	$0$	$0$	$\text{Ma1g}$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$d_{\sqrt{2}\text{xz}-\text{xy}}$	$d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)}$	$d_{\text{xz}-\sqrt{2}\text{xy}}$	$d_{y^2-x^2-\sqrt{2}\text{yz}}$	$d_{3z^2-r^2}$
$\text{s}$	$0$	$0$	$0$	$0$	$\text{Ma1g}$

==== Potential for p-f orbital mixing ====

Potential parameterized with onsite energies of irriducible representations

$A_{k,m} = \begin{cases} 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ -\frac{5 \left(\sqrt{5} \text{Ma2u1}-4 \text{Ma2u2}+4 \text{Meu1}\right)}{\sqrt{21}} & k=2\land m=0 \\ \frac{1}{2} i \left(\sqrt{30} \text{Ma2u2}-2 \sqrt{6} \text{Ma2u1}\right) & k=4\land (m=-3\lor m=3) \\ \frac{1}{2} \sqrt{\frac{3}{7}} \left(8 \sqrt{5} \text{Ma2u1}-11 \text{Ma2u2}+18 \text{Meu1}\right) & \text{True} \end{cases}$

Input format suitable for Mathematica (Quanty.nb)

Akm_D3d_Zx_A.Quanty.nb

Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {(-5*(Sqrt[5]*Ma2u1 - 4*Ma2u2 + 4*Meu1))/Sqrt[21], k == 2 && m == 0}, {(I/2)*(-2*Sqrt[6]*Ma2u1 + Sqrt[30]*Ma2u2), k == 4 && (m == -3 || m == 3)}}, (Sqrt[3/7]*(8*Sqrt[5]*Ma2u1 - 11*Ma2u2 + 18*Meu1))/2]

Input format suitable for Quanty

Akm_D3d_Zx_A.Quanty

Akm = {{2, 0, (-5)*((1/(sqrt(21)))*((sqrt(5))*(Ma2u1) + (-4)*(Ma2u2) + (4)*(Meu1)))} , 
       {4, 0, (1/2)*((sqrt(3/7))*((8)*((sqrt(5))*(Ma2u1)) + (-11)*(Ma2u2) + (18)*(Meu1)))} , 
       {4,-3, (1/2*I)*((-2)*((sqrt(6))*(Ma2u1)) + (sqrt(30))*(Ma2u2))} , 
       {4, 3, (1/2*I)*((-2)*((sqrt(6))*(Ma2u1)) + (sqrt(30))*(Ma2u2))} }

The Hamiltonian on a basis of spherical Harmonics

	${Y_{-3}^{(3)}}$	${Y_{-2}^{(3)}}$	${Y_{-1}^{(3)}}$	${Y_{0}^{(3)}}$	${Y_{1}^{(3)}}$	${Y_{2}^{(3)}}$	${Y_{3}^{(3)}}$
${Y_{-1}^{(1)}}$	$0$	$0$	$\frac{-2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}}$	$0$	$0$	$\frac{1}{6} i \left(\sqrt{30} \text{Ma2u2}-2 \sqrt{6} \text{Ma2u1}\right)$	$0$
${Y_{0}^{(1)}}$	$\frac{i \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right)}{3 \sqrt{2}}$	$0$	$0$	$\frac{1}{3} \left(\sqrt{5} \text{Ma2u1}+2 \text{Ma2u2}\right)$	$0$	$0$	$\frac{i \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right)}{3 \sqrt{2}}$
${Y_{1}^{(1)}}$	$0$	$\frac{1}{6} i \left(\sqrt{30} \text{Ma2u2}-2 \sqrt{6} \text{Ma2u1}\right)$	$0$	$0$	$\frac{-2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}}$	$0$	$0$

The Hamiltonian on a basis of symmetric functions

	$f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.}$	$f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.}$	$f_{y+\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.}$	$f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.}$	$f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.}$	$f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.}$
$p_x$	$0$	$\text{Meu1}$	$0$	$0$	$2 \text{Ma2u1}+\sqrt{5} (\text{Meu1}-\text{Ma2u2})$	$0$	$0$
$p_y$	$0$	$0$	$\text{Meu1}$	$0$	$0$	$2 \text{Ma2u1}+\sqrt{5} (\text{Meu1}-\text{Ma2u2})$	$0$
$p_z$	$\text{Ma2u1}$	$0$	$0$	$\text{Ma2u2}$	$0$	$0$	$0$

===== 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}

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