~~CLOSETOC~~
====== Orientation Zx_B ======
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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.)
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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.
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The parameterisation B of the orientation Zx is related to the orientation 0Sqrt[2]-1z of the Oh pointgroup.
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===== Symmetry Operations =====
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In the D3d Point Group, with orientation Zx_B there are the following symmetry operations
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{{:physics_chemistry:pointgroup:d3d_zx_b.png}}
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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ , |
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===== Different Settings =====
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* [[physics_chemistry:point_groups:d3d:orientation_111|Point Group D3d with orientation 111]]
* [[physics_chemistry:point_groups:d3d:orientation_zx|Point Group D3d with orientation Zx]]
* [[physics_chemistry:point_groups:d3d:orientation_zx_a|Point Group D3d with orientation Zx_A]]
* [[physics_chemistry:point_groups:d3d:orientation_zx_b|Point Group D3d with orientation Zx_B]]
* [[physics_chemistry:point_groups:d3d:orientation_z(x-y)|Point Group D3d with orientation Z(x-y)]]
* [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a|Point Group D3d with orientation Z(x-y)_A]]
* [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b|Point Group D3d with orientation Z(x-y)_B]]
* [[physics_chemistry:point_groups:d3d:orientation_zy|Point Group D3d with orientation Zy]]
* [[physics_chemistry:point_groups:d3d:orientation_zy_a|Point Group D3d with orientation Zy_A]]
* [[physics_chemistry:point_groups:d3d:orientation_zy_b|Point Group D3d with orientation Zy_B]]
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===== Character Table =====
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| $ $ ^ $ \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 $ |
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===== Product Table =====
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| $ $ ^ $ 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 $ |
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===== Sub Groups with compatible settings =====
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* [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]]
* [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]]
* [[physics_chemistry:point_groups:c3v:orientation_zx|Point Group C3v with orientation Zx]]
* [[physics_chemistry:point_groups:c3:orientation_z|Point Group C3 with orientation Z]]
* [[physics_chemistry:point_groups:ci:orientation_|Point Group Ci with orientation ]]
* [[physics_chemistry:point_groups:cs:orientation_x|Point Group Cs with orientation X]]
* [[physics_chemistry:point_groups:d3d:orientation_zx|Point Group D3d with orientation Zx]]
* [[physics_chemistry:point_groups:d3d:orientation_zx_a|Point Group D3d with orientation Zx_A]]
* [[physics_chemistry:point_groups:d3:orientation_zx|Point Group D3 with orientation Zx]]
* [[physics_chemistry:point_groups:s6:orientation_z|Point Group S6 with orientation Z]]
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===== Super Groups with compatible settings =====
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* [[physics_chemistry:point_groups:d3d:orientation_zx|Point Group D3d with orientation Zx]]
* [[physics_chemistry:point_groups:d3d:orientation_zx_a|Point Group D3d with orientation Zx_A]]
* [[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Point Group Oh with orientation 0sqrt2-1z]]
* [[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Point Group Oh with orientation 0sqrt21z]]
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===== Invariant Potential expanded on renormalized spherical Harmonics =====
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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_B the form of the expansion coefficients is:
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==== Expansion ====
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$$A_{k,m} = \begin{cases}
A(0,0) & k=0\land m=0 \\
A(2,0) & k=2\land m=0 \\
i B(4,3) & k=4\land (m=-3\lor m=3) \\
A(4,0) & k=4\land m=0 \\
A(6,6) & k=6\land (m=-6\lor m=6) \\
i B(6,3) & k=6\land (m=-3\lor m=3) \\
A(6,0) & k=6\land m=0
\end{cases}$$
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==== Input format suitable for Mathematica (Quanty.nb) ====
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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]
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==== Input format suitable for Quanty ====
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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)} }
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==== One particle coupling on a basis of spherical harmonics ====
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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 $A_{n''l'',n'l'}(k,m)$
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we can express the operator as
$$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$
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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.
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| $ $ ^ $ {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) $|
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==== Rotation matrix to symmetry adapted functions (choice is not unique) ====
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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
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| $ $ ^ $ {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_{\text{xy}+\sqrt{2}\text{xz}} $|$ 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_{-x^2+y^2-2\sqrt{2}\text{yz}} $|$ 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_{-x^2+y^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_{-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.} $|$\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_{-\left(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.\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_{\text{xy}+\sqrt{2}\text{xz}} $ ^ $ d_{-x^2+y^2-2\sqrt{2}\text{yz}} $ ^ $ d_{\text{xz}-\sqrt{2}\text{xy}} $ ^ $ d_{-x^2+y^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_{-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.} $ ^ $ 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_{-\left(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.\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_{\text{xy}+\sqrt{2}\text{xz}} $|$ 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_{-x^2+y^2-2\sqrt{2}\text{yz}} $|$ 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_{-x^2+y^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_{-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.} $|$\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_{-\left(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.\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 ====
###
$$A_{k,m} = \begin{cases}
\text{Ea1g} & k=0\land m=0 \\
0 & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{Ea1g, k == 0 && m == 0}}, 0]
###
###
Akm = {{0, 0, Ea1g} }
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^
^$ {Y_{0}^{(0)}} $|$ \text{Ea1g} $|
###
###
| $ $ ^ $ \text{s} $ ^
^$ \text{s} $|$ \text{Ea1g} $|
###
###
| $ $ ^ $ {Y_{0}^{(0)}} $ ^
^$ \text{s} $|$ 1 $|
###
###
^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_0_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: |
###
==== Potential for p orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\
\frac{5 (\text{Ea2u}-\text{Eeu})}{3} & k=2\land m=0
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(5*(Ea2u - Eeu))/3, k == 2 && m == 0}}, 0]
###
###
Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} ,
{2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} }
###
###
| $ $ ^ $ {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} $|
###
###
| $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^
^$ p_x $|$ \text{Eeu} $|$ 0 $|$ 0 $|
^$ p_y $|$ 0 $|$ \text{Eeu} $|$ 0 $|
^$ p_z $|$ 0 $|$ 0 $|$ \text{Ea2u} $|
###
###
| $ $ ^ $ {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 $|
###
###
^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_1_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: |
^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_1_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: |
^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_1_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: |
###
==== Potential for d orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\
\text{Ea1g}-\text{Eeg}\pi -2 \sqrt{2} \text{Meg} & k=2\land m=0 \\
i \sqrt{\frac{7}{5}} \left(-\sqrt{2} \text{Eeg}\pi +\sqrt{2} \text{Eeg}\sigma +\text{Meg}\right) & k=4\land (m=-3\lor m=3) \\
\frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg}\pi -7 \text{Eeg}\sigma +10 \sqrt{2} \text{Meg}\right) & k=4\land m=0
\end{cases}$$
###
###
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[7/5]*(-(Sqrt[2]*Eeg\[Pi]) + Sqrt[2]*Eeg\[Sigma] + 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]
###
###
Akm = {{0, 0, (1/5)*(Ea1g + (2)*(EegPi + EegSigma))} ,
{2, 0, Ea1g + (-1)*(EegPi) + (-2)*((sqrt(2))*(Meg))} ,
{4, 0, (1/5)*((9)*(Ea1g) + (-2)*(EegPi) + (-7)*(EegSigma) + (10)*((sqrt(2))*(Meg)))} ,
{4,-3, (I)*((sqrt(7/5))*((-1)*((sqrt(2))*(EegPi)) + (sqrt(2))*(EegSigma) + Meg))} ,
{4, 3, (I)*((sqrt(7/5))*((-1)*((sqrt(2))*(EegPi)) + (sqrt(2))*(EegSigma) + Meg))} }
###
###
| $ $ ^ $ {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{1}{3} i \left(-\sqrt{2} \text{Eeg$\pi $}+\sqrt{2} \text{Eeg$\sigma $}+\text{Meg}\right) $|$ 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{1}{3} i \left(-\sqrt{2} \text{Eeg$\pi $}+\sqrt{2} \text{Eeg$\sigma $}+\text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right) $|
###
###
| $ $ ^ $ d_{\text{xy}+\sqrt{2}\text{xz}} $ ^ $ d_{-x^2+y^2-2\sqrt{2}\text{yz}} $ ^ $ d_{\text{xz}-\sqrt{2}\text{xy}} $ ^ $ d_{-x^2+y^2+\sqrt{2}\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^
^$ d_{\text{xy}+\sqrt{2}\text{xz}} $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ 0 $|
^$ d_{-x^2+y^2-2\sqrt{2}\text{yz}} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|
^$ d_{\text{xz}-\sqrt{2}\text{xy}} $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ 0 $|
^$ d_{-x^2+y^2+\sqrt{2}\text{yz}} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|
^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1g} $|
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ d_{\text{xy}+\sqrt{2}\text{xz}} $|$ \frac{i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|
^$ d_{-x^2+y^2-2\sqrt{2}\text{yz}} $|$ -\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_{-x^2+y^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 $|
###
###
^ ^$$\text{Eeg$\sigma $}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_2_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sin (\theta ) \sin (\phi )+\sqrt{2} \cos (\theta )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(y+\sqrt{2} z\right)$$ | ::: |
^ ^$$\text{Eeg$\sigma $}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_2_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sin (\theta ) \cos (2 \phi )+2 \sqrt{2} \cos (\theta ) \sin (\phi )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(x^2-y \left(y-2 \sqrt{2} z\right)\right)$$ | ::: |
^ ^$$\text{Eeg$\pi $}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_2_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sqrt{2} \sin (\theta ) \sin (\phi )-\cos (\theta )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(\sqrt{2} y-z\right)$$ | ::: |
^ ^$$\text{Eeg$\pi $}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_2_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (\theta ) \cos (2 \phi )-2 \cos (\theta ) \sin (\phi )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(y \left(\sqrt{2} y+2 z\right)-\sqrt{2} x^2\right)$$ | ::: |
^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_2_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: |
###
==== Potential for f orbitals ====
###
$$A_{k,m} = \begin{cases}
\frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\
-\frac{5}{28} \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}-4 \sqrt{5} \text{Ma2u}+2 \sqrt{5} \text{Meu}\right) & k=2\land m=0 \\
-\frac{i \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}-\text{Ma2u}+4 \text{Meu}\right)}{\sqrt{14}} & k=4\land (m=-3\lor m=3) \\
\frac{1}{14} \left(9 \text{Ea1u}+14 \text{Ea2u1}+13 \text{Ea2u2}-34 \text{Eeu1}-2 \text{Eeu2}+4 \sqrt{5} \text{Ma2u}-16 \sqrt{5} \text{Meu}\right) & k=4\land m=0 \\
\frac{13}{60} \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) & k=6\land (m=-6\lor m=6) \\
\frac{13 i \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}-2 \text{Ma2u}-36 \text{Meu}\right)}{30 \sqrt{42}} & k=6\land (m=-3\lor m=3) \\
-\frac{13}{420} \left(3 \text{Ea1u}-32 \text{Ea2u1}-25 \text{Ea2u2}-15 \text{Eeu1}+69 \text{Eeu2}-28 \sqrt{5} \text{Ma2u}-42 \sqrt{5} \text{Meu}\right) & k=6\land m=0
\end{cases}$$
###
###
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]
###
###
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))))} }
###
###
| $ $ ^ $ {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) $|
###
###
| $ $ ^ $ 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_{-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.} $ ^ $ 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_{-\left(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.\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_{-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.} $|$ 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_{-\left(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.\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} $|
###
###
| $ $ ^ $ {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_{-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.} $|$ 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_{-\left(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.\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}} $|
###
###
^ ^$$\text{Ea2u1}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_1.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{35}{\pi }} \left(2 \sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+\cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(3 \sqrt{2} x^2 y-\sqrt{2} y^3+5 z^3-3 z\right)$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_2.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{7}{\pi }} x \left(10 \sqrt{2} y z+5 z^2-1\right)$$ | ::: |
^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_3.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \left(5 \sqrt{2} \sin (2 \theta ) \cos (2 \phi )+(5 \cos (2 \theta )+3) \sin (\phi )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} x^2 z+5 \sqrt{2} y^2 z-5 y z^2+y\right)$$ | ::: |
^ ^$$\text{Ea2u2}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_4.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+3 \cos (\theta )+5 \cos (3 \theta )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-15 \sqrt{2} x^2 y+5 \sqrt{2} y^3+4 z \left(5 z^2-3\right)\right)$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_5.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \cos (\phi ) \left(-2 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{35}{\pi }} x \left(2 \sqrt{2} y z-5 z^2+1\right)$$ | ::: |
^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_6.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \left((5 \cos (2 \theta )+3) \sin (\phi )-\sqrt{2} \sin (2 \theta ) \cos (2 \phi )\right)$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^2 z-\sqrt{2} y^2 z-5 y z^2+y\right)$$ | ::: |
^ ^$$\text{Ea1u}$$ | {{:physics_chemistry:pointgroup:d3d_zx_b_orb_3_7.png?150}} |
|$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$$ | ::: |
|$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$$ | ::: |
###
===== Coupling between two shells =====
###
Click on one of the subsections to expand it or
###
==== Potential for s-d orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & k\neq 2\lor m\neq 0 \\
\sqrt{5} \text{Ma1g} & \text{True}
\end{cases}$$
###
###
Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, Sqrt[5]*Ma1g]
###
###
Akm = {{2, 0, (sqrt(5))*(Ma1g)} }
###
###
| $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^
^$ {Y_{0}^{(0)}} $|$ 0 $|$ 0 $|$ \text{Ma1g} $|$ 0 $|$ 0 $|
###
###
| $ $ ^ $ d_{\text{xy}+\sqrt{2}\text{xz}} $ ^ $ d_{-x^2+y^2-2\sqrt{2}\text{yz}} $ ^ $ d_{\text{xz}-\sqrt{2}\text{xy}} $ ^ $ d_{-x^2+y^2+\sqrt{2}\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^
^$ \text{s} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma1g} $|
###
==== Potential for p-f orbital mixing ====
###
$$A_{k,m} = \begin{cases}
0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\
-\frac{5 \left(\sqrt{5} \text{Ma2u1}-4 \text{Ma2u2}+4 \text{Meu1}\right)}{\sqrt{21}} & k=2\land m=0 \\
i \sqrt{\frac{3}{2}} \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right) & k=4\land (m=-3\lor m=3) \\
\frac{1}{2} \sqrt{\frac{3}{7}} \left(8 \sqrt{5} \text{Ma2u1}-11 \text{Ma2u2}+18 \text{Meu1}\right) & \text{True}
\end{cases}$$
###
###
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*Sqrt[3/2]*(2*Ma2u1 - Sqrt[5]*Ma2u2), k == 4 && (m == -3 || m == 3)}}, (Sqrt[3/7]*(8*Sqrt[5]*Ma2u1 - 11*Ma2u2 + 18*Meu1))/2]
###
###
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, (I)*((sqrt(3/2))*((2)*(Ma2u1) + (-1)*((sqrt(5))*(Ma2u2))))} ,
{4, 3, (I)*((sqrt(3/2))*((2)*(Ma2u1) + (-1)*((sqrt(5))*(Ma2u2))))} }
###
###
| $ $ ^ $ {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{i \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right)}{\sqrt{6}} $|$ 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{i \left(2 \text{Ma2u1}-\sqrt{5} \text{Ma2u2}\right)}{\sqrt{6}} $|$ 0 $|$ 0 $|$ \frac{-2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}} $|$ 0 $|$ 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.} $ ^ $ f_{-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.} $ ^ $ 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_{-\left(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.\right)} $ ^ $ f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.} $ ^
^$ p_x $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ \sqrt{5} (\text{Ma2u2}-\text{Meu1})-2 \text{Ma2u1} $|$ 0 $|$ 0 $|
^$ p_y $|$ 0 $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ \sqrt{5} (\text{Ma2u2}-\text{Meu1})-2 \text{Ma2u1} $|$ 0 $|
^$ p_z $|$ \text{Ma2u1} $|$ 0 $|$ 0 $|$ \text{Ma2u2} $|$ 0 $|$ 0 $|$ 0 $|
###
===== Table of several point groups =====
###
[[physics_chemistry:point_groups|Return to Main page on Point Groups]]
###
###
^Nonaxial groups | [[physics_chemistry:point_groups:c1|C]][[physics_chemistry:point_groups:c1|1]] | [[physics_chemistry:point_groups:cs|C]][[physics_chemistry:point_groups:cs|s]] | [[physics_chemistry:point_groups:ci|C]][[physics_chemistry:point_groups:ci|i]] | | | | |
^Cn groups | [[physics_chemistry:point_groups:c2|C]][[physics_chemistry:point_groups:c2|2]] | [[physics_chemistry:point_groups:c3|C]][[physics_chemistry:point_groups:c3|3]] | [[physics_chemistry:point_groups:c4|C]][[physics_chemistry:point_groups:c4|4]] | [[physics_chemistry:point_groups:c5|C]][[physics_chemistry:point_groups:c5|5]] | [[physics_chemistry:point_groups:c6|C]][[physics_chemistry:point_groups:c6|6]] | [[physics_chemistry:point_groups:c7|C]][[physics_chemistry:point_groups:c7|7]] | [[physics_chemistry:point_groups:c8|C]][[physics_chemistry:point_groups:c8|8]] |
^Dn groups | [[physics_chemistry:point_groups:d2|D]][[physics_chemistry:point_groups:d2|2]] | [[physics_chemistry:point_groups:d3|D]][[physics_chemistry:point_groups:d3|3]] | [[physics_chemistry:point_groups:d4|D]][[physics_chemistry:point_groups:d4|4]] | [[physics_chemistry:point_groups:d5|D]][[physics_chemistry:point_groups:d5|5]] | [[physics_chemistry:point_groups:d6|D]][[physics_chemistry:point_groups:d6|6]] | [[physics_chemistry:point_groups:d7|D]][[physics_chemistry:point_groups:d7|7]] | [[physics_chemistry:point_groups:d8|D]][[physics_chemistry:point_groups:d8|8]] |
^Cnv groups | [[physics_chemistry:point_groups:c2v|C]][[physics_chemistry:point_groups:c2v|2v]] | [[physics_chemistry:point_groups:c3v|C]][[physics_chemistry:point_groups:c3v|3v]] | [[physics_chemistry:point_groups:c4v|C]][[physics_chemistry:point_groups:c4v|4v]] | [[physics_chemistry:point_groups:c5v|C]][[physics_chemistry:point_groups:c5v|5v]] | [[physics_chemistry:point_groups:c6v|C]][[physics_chemistry:point_groups:c6v|6v]] | [[physics_chemistry:point_groups:c7v|C]][[physics_chemistry:point_groups:c7v|7v]] | [[physics_chemistry:point_groups:c8v|C]][[physics_chemistry:point_groups:c8v|8v]] |
^Cnh groups | [[physics_chemistry:point_groups:c2h|C]][[physics_chemistry:point_groups:c2h|2h]] | [[physics_chemistry:point_groups:c3h|C]][[physics_chemistry:point_groups:c3h|3h]] | [[physics_chemistry:point_groups:c4h|C]][[physics_chemistry:point_groups:c4h|4h]] | [[physics_chemistry:point_groups:c5h|C]][[physics_chemistry:point_groups:c5h|5h]] | [[physics_chemistry:point_groups:c6h|C]][[physics_chemistry:point_groups:c6h|6h]] | | |
^Dnh groups | [[physics_chemistry:point_groups:d2h|D]][[physics_chemistry:point_groups:d2h|2h]] | [[physics_chemistry:point_groups:d3h|D]][[physics_chemistry:point_groups:d3h|3h]] | [[physics_chemistry:point_groups:d4h|D]][[physics_chemistry:point_groups:d4h|4h]] | [[physics_chemistry:point_groups:d5h|D]][[physics_chemistry:point_groups:d5h|5h]] | [[physics_chemistry:point_groups:d6h|D]][[physics_chemistry:point_groups:d6h|6h]] | [[physics_chemistry:point_groups:d7h|D]][[physics_chemistry:point_groups:d7h|7h]] | [[physics_chemistry:point_groups:d8h|D]][[physics_chemistry:point_groups:d8h|8h]] |
^Dnd groups | [[physics_chemistry:point_groups:d2d|D]][[physics_chemistry:point_groups:d2d|2d]] | [[physics_chemistry:point_groups:d3d|D]][[physics_chemistry:point_groups:d3d|3d]] | [[physics_chemistry:point_groups:d4d|D]][[physics_chemistry:point_groups:d4d|4d]] | [[physics_chemistry:point_groups:d5d|D]][[physics_chemistry:point_groups:d5d|5d]] | [[physics_chemistry:point_groups:d6d|D]][[physics_chemistry:point_groups:d6d|6d]] | [[physics_chemistry:point_groups:d7d|D]][[physics_chemistry:point_groups:d7d|7d]] | [[physics_chemistry:point_groups:d8d|D]][[physics_chemistry:point_groups:d8d|8d]] |
^Sn groups | [[physics_chemistry:point_groups:S2|S]][[physics_chemistry:point_groups:S2|2]] | [[physics_chemistry:point_groups:S4|S]][[physics_chemistry:point_groups:S4|4]] | [[physics_chemistry:point_groups:S6|S]][[physics_chemistry:point_groups:S6|6]] | [[physics_chemistry:point_groups:S8|S]][[physics_chemistry:point_groups:S8|8]] | [[physics_chemistry:point_groups:S10|S]][[physics_chemistry:point_groups:S10|10]] | [[physics_chemistry:point_groups:S12|S]][[physics_chemistry:point_groups:S12|12]] | |
^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]][[physics_chemistry:point_groups:Th|h]] | [[physics_chemistry:point_groups:Td|T]][[physics_chemistry:point_groups:Td|d]] | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]][[physics_chemistry:point_groups:Oh|h]] | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]][[physics_chemistry:point_groups:Ih|h]] |
^Linear groups | [[physics_chemistry:point_groups:cinfv|C]][[physics_chemistry:point_groups:cinfv|$\infty$v]] | [[physics_chemistry:point_groups:cinfv|D]][[physics_chemistry:point_groups:dinfh|$\infty$h]] | | | | | |
###