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 — physics_chemistry:point_groups:d3d:orientation_z_x-y_a [2018/09/06 14:00] (current)Maurits W. Haverkort created 2018/09/06 14:00 Maurits W. Haverkort created 2018/09/06 14:00 Maurits W. Haverkort created Line 1: Line 1: + ~~CLOSETOC~~ + ====== Orientation Z(x-y)_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 parameterization A of the orientation Z(x-y) is related to the orientation 111z of the Oh pointgroup. + + ### + + ===== Symmetry Operations ===== + + ### + + In the D3d Point Group, with orientation Z(x-y)_A there are the following symmetry operations + + ### + + ### + + {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a.png}} + + ### + + ### + + ^ Operator ^ Orientation ^ + ^ $\text{E}$ | $\{0,0,0\}$ , | + ^ $C_3$ | $\{0,0,1\}$ , $\{0,​0,​-1\}$ , | + ^ $C_2$ | $\{1,​-1,​0\}$ , $\left\{2+\sqrt{3},​1,​0\right\}$ , $\left\{1,​2+\sqrt{3},​0\right\}$ , | + ^ $\text{i}$ | $\{0,0,0\}$ , | + ^ $S_6$ | $\{0,0,1\}$ , $\{0,​0,​-1\}$ , | + ^ $\sigma _d$ | $\{1,​-1,​0\}$ , $\left\{2+\sqrt{3},​1,​0\right\}$ , $\left\{1,​2+\sqrt{3},​0\right\}$ , | + + ### + + ===== Different Settings ===== + + ### + + * [[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]] + + ### + + ===== 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 ===== + + ### + + * [[physics_chemistry:​point_groups:​c1:​orientation_1|Point Group C1 with orientation 1]] + * [[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:​d3d:​orientation_z(x-y)|Point Group D3d with orientation Z(x-y)]] + * [[physics_chemistry:​point_groups:​d3d:​orientation_z(x-y)_b|Point Group D3d with orientation Z(x-y)_B]] + * [[physics_chemistry:​point_groups:​s6:​orientation_z|Point Group S6 with orientation Z]] + + ### + + ===== Super Groups with compatible settings ===== + + ### + + * [[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)_b|Point Group D3d with orientation Z(x-y)_B]] + * [[physics_chemistry:​point_groups:​oh:​orientation_11-1z|Point Group Oh with orientation 11-1z]] + * [[physics_chemistry:​point_groups:​oh:​orientation_111z|Point Group Oh with orientation 111z]] + + ### + + ===== 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 Z(x-y)_A the form of the expansion coefficients is: + + ### + + ==== Expansion ==== + + ### + + ​$$A_{k,​m} = \begin{cases} + ​A(0,​0) & k=0\land m=0 \\ + ​A(2,​0) & k=2\land m=0 \\ + ​(-1+i) A(4,3) & k=4\land m=-3 \\ + ​A(4,​0) & k=4\land m=0 \\ + (1+i) A(4,3) & k=4\land m=3 \\ + -i B(6,6) & k=6\land m=-6 \\ + ​(-1+i) A(6,3) & k=6\land m=-3 \\ + ​A(6,​0) & k=6\land m=0 \\ + (1+i) A(6,3) & k=6\land m=3 \\ + i B(6,6) & k=6\land m=6 + \end{cases}$$ + + ### + + ==== Input format suitable for Mathematica (Quanty.nb) ==== + + ### + + ​ + + Akm[k_,​m_]:​=Piecewise[{{A[0,​ 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {(-1 + I)*A[4, 3], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {(1 + I)*A[4, 3], k == 4 && m == 3}, {(-I)*B[6, 6], k == 6 && m == -6}, {(-1 + I)*A[6, 3], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {(1 + I)*A[6, 3], k == 6 && m == 3}, {I*B[6, 6], k == 6 && m == 6}}, 0] + + ​ + + ### + + ==== Input format suitable for Quanty ==== + + ### + + ​ + + Akm = {{0, 0, A(0,0)} , + {2, 0, A(2,0)} , + {4, 0, A(4,0)} , + ​{4,​-3,​ (-1+1*I)*(A(4,​3))} , + {4, 3, (1+1*I)*(A(4,​3))} , + {6, 0, A(6,0)} , + ​{6,​-3,​ (-1+1*I)*(A(6,​3))} , + {6, 3, (1+1*I)*(A(6,​3))} , + ​{6,​-6,​ (-I)*(B(6,​6))} , + {6, 6, (I)*(B(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. + + ### + + + + ### + + |    ^  ${Y_{0}^{(0)}}$  ^  ${Y_{-1}^{(1)}}$  ^  ${Y_{0}^{(1)}}$  ^  ${Y_{1}^{(1)}}$  ^  ${Y_{-2}^{(2)}}$  ^  ${Y_{-1}^{(2)}}$  ^  ${Y_{0}^{(2)}}$  ^  ${Y_{1}^{(2)}}$  ^  ${Y_{2}^{(2)}}$  ^  ${Y_{-3}^{(3)}}$  ^  ${Y_{-2}^{(3)}}$  ^  ${Y_{-1}^{(3)}}$  ^  ${Y_{0}^{(3)}}$  ^  ${Y_{1}^{(3)}}$  ^  ${Y_{2}^{(3)}}$  ^  ${Y_{3}^{(3)}}$  ^ + ^${Y_{0}^{(0)}}$|$\text{Ass}(0,​0)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$\frac{\text{Asd}(2,​0)}{\sqrt{5}}$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^${Y_{-1}^{(1)}}$|$\color{darkred}{ 0 }$|$\text{App}(0,​0)-\frac{1}{5} \text{App}(2,​0)$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0)$|$0$|$0$|$\left(-\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,​3)$|$0$| + ^${Y_{0}^{(1)}}$|$\color{darkred}{ 0 }$|$0$|$\text{App}(0,​0)+\frac{2}{5} \text{App}(2,​0)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$-\frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,​3)}{\sqrt{3}}$|$0$|$0$|$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}$|$0$|$0$|$\frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,​3)}{\sqrt{3}}$| + ^${Y_{1}^{(1)}}$|$\color{darkred}{ 0 }$|$0$|$0$|$\text{App}(0,​0)-\frac{1}{5} \text{App}(2,​0)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,​3)$|$0$|$0$|$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0)$|$0$|$0$| + ^${Y_{-2}^{(2)}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\text{Add}(0,​0)-\frac{2}{7} \text{Add}(2,​0)+\frac{1}{21} \text{Add}(4,​0)$|$0$|$0$|$\left(\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^${Y_{-1}^{(2)}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\text{Add}(0,​0)+\frac{1}{7} \text{Add}(2,​0)-\frac{4}{21} \text{Add}(4,​0)$|$0$|$0$|$\left(-\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^${Y_{0}^{(2)}}$|$\frac{\text{Asd}(2,​0)}{\sqrt{5}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$\text{Add}(0,​0)+\frac{2}{7} \text{Add}(2,​0)+\frac{2}{7} \text{Add}(4,​0)$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^${Y_{1}^{(2)}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\left(\frac{1}{3}+\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$0$|$0$|$\text{Add}(0,​0)+\frac{1}{7} \text{Add}(2,​0)-\frac{4}{21} \text{Add}(4,​0)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^${Y_{2}^{(2)}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\left(-\frac{1}{3}-\frac{i}{3}\right) \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$0$|$0$|$\text{Add}(0,​0)-\frac{2}{7} \text{Add}(2,​0)+\frac{1}{21} \text{Add}(4,​0)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^${Y_{-3}^{(3)}}$|$\color{darkred}{ 0 }$|$0$|$-\frac{\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,​3)}{\sqrt{3}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\text{Aff}(0,​0)-\frac{1}{3} \text{Aff}(2,​0)+\frac{1}{11} \text{Aff}(4,​0)-\frac{5}{429} \text{Aff}(6,​0)$|$0$|$0$|$\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,​3)-\left(\frac{10}{143}-\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,​3)$|$0$|$0$|$\frac{10}{13} i \sqrt{\frac{7}{33}} \text{Bff}(6,​6)$| + ^${Y_{-2}^{(3)}}$|$\color{darkred}{ 0 }$|$0$|$0$|$\left(\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,​3)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\text{Aff}(0,​0)-\frac{7}{33} \text{Aff}(4,​0)+\frac{10}{143} \text{Aff}(6,​0)$|$0$|$0$|$\left(\frac{1}{33}-\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,​3)+\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(6,​3)$|$0$|$0$| + ^${Y_{-1}^{(3)}}$|$\color{darkred}{ 0 }$|$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0)$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$\text{Aff}(0,​0)+\frac{1}{5} \text{Aff}(2,​0)+\frac{1}{33} \text{Aff}(4,​0)-\frac{25}{143} \text{Aff}(6,​0)$|$0$|$0$|$\left(-\frac{1}{33}+\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,​3)-\left(\frac{5}{143}-\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(6,​3)$|$0$| + ^${Y_{0}^{(3)}}$|$\color{darkred}{ 0 }$|$0$|$\frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{4 \text{Apf}(4,​0)}{3 \sqrt{21}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,​3)-\left(\frac{10}{143}+\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,​3)$|$0$|$0$|$\text{Aff}(0,​0)+\frac{4}{15} \text{Aff}(2,​0)+\frac{2}{11} \text{Aff}(4,​0)+\frac{100}{429} \text{Aff}(6,​0)$|$0$|$0$|$\left(\frac{10}{143}-\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,​3)-\left(\frac{1}{11}-\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,​3)$| + ^${Y_{1}^{(3)}}$|$\color{darkred}{ 0 }$|$0$|$0$|$\frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,​0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,​0)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\left(\frac{1}{33}+\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,​3)+\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(6,​3)$|$0$|$0$|$\text{Aff}(0,​0)+\frac{1}{5} \text{Aff}(2,​0)+\frac{1}{33} \text{Aff}(4,​0)-\frac{25}{143} \text{Aff}(6,​0)$|$0$|$0$| + ^${Y_{2}^{(3)}}$|$\color{darkred}{ 0 }$|$\left(-\frac{1}{3}-\frac{i}{3}\right) \text{Apf}(4,​3)$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$\left(-\frac{1}{33}-\frac{i}{33}\right) \sqrt{14} \text{Aff}(4,​3)-\left(\frac{5}{143}+\frac{5 i}{143}\right) \sqrt{42} \text{Aff}(6,​3)$|$0$|$0$|$\text{Aff}(0,​0)-\frac{7}{33} \text{Aff}(4,​0)+\frac{10}{143} \text{Aff}(6,​0)$|$0$| + ^${Y_{3}^{(3)}}$|$\color{darkred}{ 0 }$|$0$|$\frac{\left(\frac{1}{3}+\frac{i}{3}\right) \text{Apf}(4,​3)}{\sqrt{3}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$-\frac{10}{13} i \sqrt{\frac{7}{33}} \text{Bff}(6,​6)$|$0$|$0$|$\left(\frac{10}{143}+\frac{10 i}{143}\right) \sqrt{\frac{7}{3}} \text{Aff}(6,​3)-\left(\frac{1}{11}+\frac{i}{11}\right) \sqrt{7} \text{Aff}(4,​3)$|$0$|$0$|$\text{Aff}(0,​0)-\frac{1}{3} \text{Aff}(2,​0)+\frac{1}{11} \text{Aff}(4,​0)-\frac{5}{429} \text{Aff}(6,​0)$| + + + ### + + ==== 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 + + ### + + + + ### + + |    ^  ${Y_{0}^{(0)}}$  ^  ${Y_{-1}^{(1)}}$  ^  ${Y_{0}^{(1)}}$  ^  ${Y_{1}^{(1)}}$  ^  ${Y_{-2}^{(2)}}$  ^  ${Y_{-1}^{(2)}}$  ^  ${Y_{0}^{(2)}}$  ^  ${Y_{1}^{(2)}}$  ^  ${Y_{2}^{(2)}}$  ^  ${Y_{-3}^{(3)}}$  ^  ${Y_{-2}^{(3)}}$  ^  ${Y_{-1}^{(3)}}$  ^  ${Y_{0}^{(3)}}$  ^  ${Y_{1}^{(3)}}$  ^  ${Y_{2}^{(3)}}$  ^  ${Y_{3}^{(3)}}$  ^ + ^$\text{s}$|$1$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$p_z$|$\color{darkred}{ 0 }$|$0$|$1$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$0$|$0$|$0$| + ^$p_x$|$\color{darkred}{ 0 }$|$\frac{1}{\sqrt{2}}$|$0$|$-\frac{1}{\sqrt{2}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$0$|$0$|$0$| + ^$p_y$|$\color{darkred}{ 0 }$|$\frac{i}{\sqrt{2}}$|$0$|$\frac{i}{\sqrt{2}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$0$|$0$|$0$| + ^$d_{3z^2-r^2}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$1$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{(x-y)(x+y+z)}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{1}{\sqrt{3}}$|$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$|$0$|$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$|$\frac{1}{\sqrt{3}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{2\text{xy}-\text{xz}-\text{yz}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{i}{\sqrt{3}}$|$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$|$0$|$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$|$-\frac{i}{\sqrt{3}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{(x-y)(x+y-2z)}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{1}{\sqrt{6}}$|$-\frac{1-i}{\sqrt{6}}$|$0$|$\frac{1+i}{\sqrt{6}}$|$\frac{1}{\sqrt{6}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{\text{yz}+\text{xz}+\text{xy}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{i}{\sqrt{6}}$|$\frac{1+i}{\sqrt{6}}$|$0$|$-\frac{1-i}{\sqrt{6}}$|$-\frac{i}{\sqrt{6}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{1}{2}+\frac{i}{2}$|$0$|$0$|$0$|$0$|$0$|$-\frac{1}{2}+\frac{i}{2}$| + ^$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{1}{3}-\frac{i}{3}$|$0$|$0$|$-\frac{\sqrt{5}}{3}$|$0$|$0$|$-\frac{1}{3}-\frac{i}{3}$| + ^$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\left(\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$|$0$|$0$|$\frac{2}{3}$|$0$|$0$|$\left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$| + ^$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$-\frac{1}{2 \sqrt{3}}$|$0$|$\frac{1}{2 \sqrt{3}}$|$\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$0$| + ^$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$-\frac{i}{2 \sqrt{3}}$|$0$|$-\frac{i}{2 \sqrt{3}}$|$\left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$0$| + ^$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$-\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$|$-\frac{\sqrt{\frac{5}{3}}}{2}$|$0$|$\frac{\sqrt{\frac{5}{3}}}{2}$|$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$|$0$| + ^$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$|$-\frac{1}{2} i \sqrt{\frac{5}{3}}$|$0$|$-\frac{1}{2} i \sqrt{\frac{5}{3}}$|$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$|$0$| + + + ### + + ==== One particle coupling on a basis of symmetry adapted functions ==== + + ### + + After rotation we find + + ### + + + + ### + + |    ^  $\text{s}$  ^  $p_z$  ^  $p_x$  ^  $p_y$  ^  $d_{3z^2-r^2}$  ^  $d_{(x-y)(x+y+z)}$  ^  $d_{2\text{xy}-\text{xz}-\text{yz}}$  ^  $d_{(x-y)(x+y-2z)}$  ^  $d_{\text{yz}+\text{xz}+\text{xy}}$  ^  $f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$  ^  $f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$  ^  $f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$  ^  $f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^ + ^$\text{s}$|$\text{Ass}(0,​0)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\frac{\text{Asd}(2,​0)}{\sqrt{5}}$|$0$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$p_z$|$\color{darkred}{ 0 }$|$\text{App}(0,​0)+\frac{2}{5} \text{App}(2,​0)$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$-\sqrt{\frac{3}{35}} \text{Apf}(2,​0)-\frac{4}{9} \sqrt{\frac{5}{21}} \text{Apf}(4,​0)-\frac{4 \text{Apf}(4,​3)}{9 \sqrt{3}}$|$\frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{8 \text{Apf}(4,​0)}{9 \sqrt{21}}-\frac{2}{9} \sqrt{\frac{5}{3}} \text{Apf}(4,​3)$|$0$|$0$|$0$|$0$| + ^$p_x$|$\color{darkred}{ 0 }$|$0$|$\text{App}(0,​0)-\frac{1}{5} \text{App}(2,​0)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{\text{Apf}(4,​0)}{3 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,​3)$|$0$|$-\sqrt{\frac{3}{35}} \text{Apf}(2,​0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​0)+\frac{\text{Apf}(4,​3)}{3 \sqrt{3}}$|$0$| + ^$p_y$|$\color{darkred}{ 0 }$|$0$|$0$|$\text{App}(0,​0)-\frac{1}{5} \text{App}(2,​0)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{\text{Apf}(4,​0)}{3 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,​3)$|$0$|$-\sqrt{\frac{3}{35}} \text{Apf}(2,​0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​0)+\frac{\text{Apf}(4,​3)}{3 \sqrt{3}}$| + ^$d_{3z^2-r^2}$|$\frac{\text{Asd}(2,​0)}{\sqrt{5}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\text{Add}(0,​0)+\frac{2}{7} \text{Add}(2,​0)+\frac{2}{7} \text{Add}(4,​0)$|$0$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{(x-y)(x+y+z)}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\text{Add}(0,​0)-\frac{1}{7} \text{Add}(2,​0)-\frac{2}{63} \text{Add}(4,​0)-\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$0$|$-\frac{1}{7} \sqrt{2} \text{Add}(2,​0)+\frac{5}{63} \sqrt{2} \text{Add}(4,​0)+\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,​3)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{2\text{xy}-\text{xz}-\text{yz}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$\text{Add}(0,​0)-\frac{1}{7} \text{Add}(2,​0)-\frac{2}{63} \text{Add}(4,​0)-\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$0$|$-\frac{1}{7} \sqrt{2} \text{Add}(2,​0)+\frac{5}{63} \sqrt{2} \text{Add}(4,​0)+\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,​3)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{(x-y)(x+y-2z)}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$-\frac{1}{7} \sqrt{2} \text{Add}(2,​0)+\frac{5}{63} \sqrt{2} \text{Add}(4,​0)+\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,​3)$|$0$|$\text{Add}(0,​0)-\frac{1}{9} \text{Add}(4,​0)+\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$d_{\text{yz}+\text{xz}+\text{xy}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$-\frac{1}{7} \sqrt{2} \text{Add}(2,​0)+\frac{5}{63} \sqrt{2} \text{Add}(4,​0)+\frac{1}{9} \sqrt{\frac{10}{7}} \text{Add}(4,​3)$|$0$|$\text{Add}(0,​0)-\frac{1}{9} \text{Add}(4,​0)+\frac{4}{9} \sqrt{\frac{5}{7}} \text{Add}(4,​3)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| + ^$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\text{Aff}(0,​0)-\frac{1}{3} \text{Aff}(2,​0)+\frac{1}{11} \text{Aff}(4,​0)-\frac{5}{429} \text{Aff}(6,​0)-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,​6)$|$0$|$0$|$0$|$0$|$0$|$0$| + ^$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$|$\color{darkred}{ 0 }$|$-\sqrt{\frac{3}{35}} \text{Apf}(2,​0)-\frac{4}{9} \sqrt{\frac{5}{21}} \text{Apf}(4,​0)-\frac{4 \text{Apf}(4,​3)}{9 \sqrt{3}}$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$\text{Aff}(0,​0)+\frac{14}{99} \text{Aff}(4,​0)-\frac{8}{99} \sqrt{35} \text{Aff}(4,​3)+\frac{160 \text{Aff}(6,​0)}{1287}+\frac{80 \sqrt{\frac{35}{3}} \text{Aff}(6,​3)}{1287}+\frac{40}{117} \sqrt{\frac{7}{33}} \text{Bff}(6,​6)$|$-\frac{2 \text{Aff}(2,​0)}{3 \sqrt{5}}-\frac{2}{99} \sqrt{5} \text{Aff}(4,​0)-\frac{2}{99} \sqrt{7} \text{Aff}(4,​3)-\frac{70 \sqrt{5} \text{Aff}(6,​0)}{1287}+\frac{20 \sqrt{\frac{7}{3}} \text{Aff}(6,​3)}{1287}+\frac{20}{117} \sqrt{\frac{35}{33}} \text{Bff}(6,​6)$|$0$|$0$|$0$|$0$| + ^$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$|$\color{darkred}{ 0 }$|$\frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{8 \text{Apf}(4,​0)}{9 \sqrt{21}}-\frac{2}{9} \sqrt{\frac{5}{3}} \text{Apf}(4,​3)$|$0$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$-\frac{2 \text{Aff}(2,​0)}{3 \sqrt{5}}-\frac{2}{99} \sqrt{5} \text{Aff}(4,​0)-\frac{2}{99} \sqrt{7} \text{Aff}(4,​3)-\frac{70 \sqrt{5} \text{Aff}(6,​0)}{1287}+\frac{20 \sqrt{\frac{7}{3}} \text{Aff}(6,​3)}{1287}+\frac{20}{117} \sqrt{\frac{35}{33}} \text{Bff}(6,​6)$|$\text{Aff}(0,​0)-\frac{1}{15} \text{Aff}(2,​0)+\frac{13}{99} \text{Aff}(4,​0)+\frac{8}{99} \sqrt{35} \text{Aff}(4,​3)+\frac{125 \text{Aff}(6,​0)}{1287}-\frac{80 \sqrt{\frac{35}{3}} \text{Aff}(6,​3)}{1287}+\frac{50}{117} \sqrt{\frac{7}{33}} \text{Bff}(6,​6)$|$0$|$0$|$0$|$0$| + ^$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{\text{Apf}(4,​0)}{3 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,​3)$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\text{Aff}(0,​0)+\frac{1}{30} \text{Aff}(2,​0)-\frac{17}{99} \text{Aff}(4,​0)+\frac{2}{99} \sqrt{35} \text{Aff}(4,​3)+\frac{25}{858} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,​3)$|$0$|$\frac{\text{Aff}(2,​0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,​0)+\frac{4}{99} \sqrt{7} \text{Aff}(4,​3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3)$|$0$| + ^$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$-\frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,​0)+\frac{\text{Apf}(4,​0)}{3 \sqrt{21}}-\frac{1}{3} \sqrt{\frac{5}{3}} \text{Apf}(4,​3)$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$\text{Aff}(0,​0)+\frac{1}{30} \text{Aff}(2,​0)-\frac{17}{99} \text{Aff}(4,​0)+\frac{2}{99} \sqrt{35} \text{Aff}(4,​3)+\frac{25}{858} \text{Aff}(6,​0)+\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,​3)$|$0$|$\frac{\text{Aff}(2,​0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,​0)+\frac{4}{99} \sqrt{7} \text{Aff}(4,​3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3)$| + ^$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$-\sqrt{\frac{3}{35}} \text{Apf}(2,​0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​0)+\frac{\text{Apf}(4,​3)}{3 \sqrt{3}}$|$0$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$\frac{\text{Aff}(2,​0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,​0)+\frac{4}{99} \sqrt{7} \text{Aff}(4,​3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3)$|$0$|$\text{Aff}(0,​0)+\frac{1}{6} \text{Aff}(2,​0)-\frac{1}{99} \text{Aff}(4,​0)-\frac{2}{99} \sqrt{35} \text{Aff}(4,​3)-\frac{115}{858} \text{Aff}(6,​0)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,​3)$|$0$| + ^$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$\color{darkred}{ 0 }$|$0$|$0$|$-\sqrt{\frac{3}{35}} \text{Apf}(2,​0)+\frac{1}{3} \sqrt{\frac{5}{21}} \text{Apf}(4,​0)+\frac{\text{Apf}(4,​3)}{3 \sqrt{3}}$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$0$|$0$|$0$|$0$|$\frac{\text{Aff}(2,​0)}{6 \sqrt{5}}+\frac{4}{99} \sqrt{5} \text{Aff}(4,​0)+\frac{4}{99} \sqrt{7} \text{Aff}(4,​3)-\frac{35}{858} \sqrt{5} \text{Aff}(6,​0)+\frac{20}{143} \sqrt{\frac{7}{3}} \text{Aff}(6,​3)$|$0$|$\text{Aff}(0,​0)+\frac{1}{6} \text{Aff}(2,​0)-\frac{1}{99} \text{Aff}(4,​0)-\frac{2}{99} \sqrt{35} \text{Aff}(4,​3)-\frac{115}{858} \text{Aff}(6,​0)-\frac{10}{143} \sqrt{\frac{35}{3}} \text{Aff}(6,​3)$| + + + ### + + ===== 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 <​hiddenSwitch expand all> ​ + + ### + + ==== 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] + + ​ + + ### + + <​hidden **Input format suitable for Quanty** > + + ### + + ​ + + 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_z(x-y)_a_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] + + ​ + + ### + + <​hidden **Input format suitable for Quanty** > + + ### + + ​ + + Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , + {2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} } + + ​ + + ### + + ​ + + + ### + + |    ^  ${Y_{-1}^{(1)}}$  ^  ${Y_{0}^{(1)}}$  ^  ${Y_{1}^{(1)}}$  ^ + ^${Y_{-1}^{(1)}}$|$\text{Eeu}$|$0$|$0$| + ^${Y_{0}^{(1)}}$|$0$|$\text{Ea2u}$|$0$| + ^${Y_{1}^{(1)}}$|$0$|$0$|$\text{Eeu}$| + + + ### + + ​ + + + ### + + |    ^  $p_z$  ^  $p_x$  ^  $p_y$  ^ + ^$p_z$|$\text{Ea2u}$|$0$|$0$| + ^$p_x$|$0$|$\text{Eeu}$|$0$| + ^$p_y$|$0$|$0$|$\text{Eeu}$| + + + ### + + ​ + + + ### + + |    ^  ${Y_{-1}^{(1)}}$  ^  ${Y_{0}^{(1)}}$  ^  ${Y_{1}^{(1)}}$  ^ + ^$p_z$|$0$|$1$|$0$| + ^$p_x$|$\frac{1}{\sqrt{2}}$|$0$|$-\frac{1}{\sqrt{2}}$| + ^$p_y$|$\frac{i}{\sqrt{2}}$|$0$|$\frac{i}{\sqrt{2}}$| + + + ### + + ​ + + + ### + + ^ ^$$\text{Ea2u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_1_1.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | + ^ ^$$\text{Eeu}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_1_2.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_z(x-y)_a_orb_1_3.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | + + + ### + + ​ + ==== Potential for d orbitals ==== + + + + ### + + ​$$A_{k,​m} = \begin{cases} + ​\frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\ + ​\text{Ea1g}-\text{Eeg}\pi -2 \sqrt{2} \text{Meg} & k=2\land m=0 \\ + ​\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{7}{5}} \left(2 \text{Eeg}\pi -2 \text{Eeg}\sigma -\sqrt{2} \text{Meg}\right) & k=4\land m=-3 \\ + ​\frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg}\pi -7 \text{Eeg}\sigma +10 \sqrt{2} \text{Meg}\right) & k=4\land m=0 \\ + ​\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{7}{5}} \left(2 \text{Eeg}\pi -2 \text{Eeg}\sigma -\sqrt{2} \text{Meg}\right) & k=4\land m=3 + \end{cases}$$ + + ### + + ​ + + + ### + + ​ + + 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}, {(1/2 - I/​2)*Sqrt[7/​5]*(2*Eeg\[Pi] - 2*Eeg\[Sigma] - Sqrt[2]*Meg),​ k == 4 && m == -3}, {(9*Ea1g - 2*Eeg\[Pi] - 7*Eeg\[Sigma] + 10*Sqrt[2]*Meg)/​5,​ k == 4 && m == 0}, {(-1/2 - I/​2)*Sqrt[7/​5]*(2*Eeg\[Pi] - 2*Eeg\[Sigma] - Sqrt[2]*Meg),​ k == 4 && m == 3}}, 0] + + ​ + + ### + + <​hidden **Input format suitable for Quanty** > + + ### + + ​ + + 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, (-1/​2+-1/​2*I)*((sqrt(7/​5))*((2)*(EegPi) + (-2)*(EegSigma) + (-1)*((sqrt(2))*(Meg))))} , + ​{4,​-3,​ (1/​2+-1/​2*I)*((sqrt(7/​5))*((2)*(EegPi) + (-2)*(EegSigma) + (-1)*((sqrt(2))*(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 $}+2 \sqrt{2} \text{Meg}\right)$|$0$|$0$|$\left(\frac{1}{6}-\frac{i}{6}\right) \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right)$|$0$| + ^${Y_{-1}^{(2)}}$|$0$|$\frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right)$|$0$|$0$|$\left(-\frac{1}{6}+\frac{i}{6}\right) \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right)$| + ^${Y_{0}^{(2)}}$|$0$|$0$|$\text{Ea1g}$|$0$|$0$| + ^${Y_{1}^{(2)}}$|$\left(\frac{1}{6}+\frac{i}{6}\right) \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma $}+\sqrt{2} \text{Meg}\right)$|$0$|$0$|$\frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right)$|$0$| + ^${Y_{2}^{(2)}}$|$0$|$\left(\frac{1}{6}+\frac{i}{6}\right) \left(2 \text{Eeg$\pi $}-2 \text{Eeg$\sigma $}-\sqrt{2} \text{Meg}\right)$|$0$|$0$|$\frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right)$| + + + ### + + ​ + + + ### + + |    ^  $d_{3z^2-r^2}$  ^  $d_{(x-y)(x+y+z)}$  ^  $d_{2\text{xy}-\text{xz}-\text{yz}}$  ^  $d_{(x-y)(x+y-2z)}$  ^  $d_{\text{yz}+\text{xz}+\text{xy}}$  ^ + ^$d_{3z^2-r^2}$|$\text{Ea1g}$|$0$|$0$|$0$|$0$| + ^$d_{(x-y)(x+y+z)}$|$0$|$\text{Eeg$\pi $}$|$0$|$\text{Meg}$|$0$| + ^$d_{2\text{xy}-\text{xz}-\text{yz}}$|$0$|$0$|$\text{Eeg$\pi $}$|$0$|$\text{Meg}$| + ^$d_{(x-y)(x+y-2z)}$|$0$|$\text{Meg}$|$0$|$\text{Eeg$\sigma $}$|$0$| + ^$d_{\text{yz}+\text{xz}+\text{xy}}$|$0$|$0$|$\text{Meg}$|$0$|$\text{Eeg$\sigma $}$| + + + ### + + ​ + + + ### + + |    ^  ${Y_{-2}^{(2)}}$  ^  ${Y_{-1}^{(2)}}$  ^  ${Y_{0}^{(2)}}$  ^  ${Y_{1}^{(2)}}$  ^  ${Y_{2}^{(2)}}$  ^ + ^$d_{3z^2-r^2}$|$0$|$0$|$1$|$0$|$0$| + ^$d_{(x-y)(x+y+z)}$|$\frac{1}{\sqrt{3}}$|$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$|$0$|$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$|$\frac{1}{\sqrt{3}}$| + ^$d_{2\text{xy}-\text{xz}-\text{yz}}$|$\frac{i}{\sqrt{3}}$|$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{3}}$|$0$|$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{3}}$|$-\frac{i}{\sqrt{3}}$| + ^$d_{(x-y)(x+y-2z)}$|$\frac{1}{\sqrt{6}}$|$-\frac{1-i}{\sqrt{6}}$|$0$|$\frac{1+i}{\sqrt{6}}$|$\frac{1}{\sqrt{6}}$| + ^$d_{\text{yz}+\text{xz}+\text{xy}}$|$\frac{i}{\sqrt{6}}$|$\frac{1+i}{\sqrt{6}}$|$0$|$-\frac{1-i}{\sqrt{6}}$|$-\frac{i}{\sqrt{6}}$| + + + ### + + ​ + + + ### + + ^ ^$$\text{Ea1g}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_1.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | + ^ ^$$\text{Eeg\pi }$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_2.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta ))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (x-y) (x+y+z)$$ | ::: | + ^ ^$$\text{Eeg\pi }$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_3.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} \sin (\theta ) (\sin (\theta ) \sin (2 \phi )-\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{2 \pi }} (2 x y-x z-y z)$$ | ::: | + ^ ^$$\text{Eeg\sigma }$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_4.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\cos (\phi )-\sin (\phi )) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))-2 \cos (\theta ))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} (x-y) (x+y-2 z)$$ | ::: | + ^ ^$$\text{Eeg\sigma }$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_2_5.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) (\sin (\theta ) \sin (\phi ) \cos (\phi )+\cos (\theta ) (\sin (\phi )+\cos (\phi )))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} (x (y+z)+y z)$$ | ::: | + + + ### + + ​ + ==== Potential for f orbitals ==== + + + + ### + + ​$$A_{k,​m} = \begin{cases} + ​\frac{1}{7} (\text{Ea1u}+\text{Ea2u1}+\text{Ea2u2}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\ + ​-\frac{5}{28} \left(5 \text{Ea1u}+\text{Ea2u2}-\text{Eeu1}-5 \text{Eeu2}+4 \sqrt{5} \text{Ma2u}-2 \sqrt{5} \text{Meu}\right) & k=2\land m=0 \\ + ​\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}+\text{Ma2u}-4 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=-3 \\ + ​\frac{1}{14} \left(9 \text{Ea1u}+14 \text{Ea2u1}+13 \text{Ea2u2}-34 \text{Eeu1}-2 \text{Eeu2}-4 \sqrt{5} \text{Ma2u}+16 \sqrt{5} \text{Meu}\right) & k=4\land m=0 \\ + ​-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}+\text{Ma2u}-4 \text{Meu}\right)}{\sqrt{7}} & k=4\land m=3 \\ + ​\frac{13}{60} i \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) & k=6\land m=-6 \\ + ​-\frac{\left(\frac{13}{60}-\frac{13 i}{60}\right) \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}+2 \text{Ma2u}+36 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=-3 \\ + ​-\frac{13}{420} \left(3 \text{Ea1u}-32 \text{Ea2u1}-25 \text{Ea2u2}-15 \text{Eeu1}+69 \text{Eeu2}+28 \sqrt{5} \text{Ma2u}+42 \sqrt{5} \text{Meu}\right) & k=6\land m=0 \\ + ​\frac{\left(\frac{13}{60}+\frac{13 i}{60}\right) \left(4 \sqrt{5} \text{Ea2u1}-4 \sqrt{5} \text{Ea2u2}+9 \sqrt{5} \text{Eeu1}-9 \sqrt{5} \text{Eeu2}+2 \text{Ma2u}+36 \text{Meu}\right)}{\sqrt{21}} & k=6\land m=3 \\ + ​-\frac{13}{60} i \sqrt{\frac{11}{21}} \left(9 \text{Ea1u}-4 \text{Ea2u1}-5 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) & k=6\land m=6 + \end{cases}$$ + + ### + + ​ + + + ### + + ​ + + 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}, {((1/2 - I/​2)*(2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 + Ma2u - 4*Meu))/​Sqrt[7],​ k == 4 && 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}, {((-1/2 - I/​2)*(2*Sqrt[5]*Ea2u1 - 2*Sqrt[5]*Ea2u2 - Sqrt[5]*Eeu1 + Sqrt[5]*Eeu2 + Ma2u - 4*Meu))/​Sqrt[7],​ k == 4 && m == 3}, {((13*I)/​60)*Sqrt[11/​21]*(9*Ea1u - 4*Ea2u1 - 5*Ea2u2 - 4*Sqrt[5]*Ma2u),​ k == 6 && m == -6}, {((-13/60 + (13*I)/​60)*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 + 2*Ma2u + 36*Meu))/​Sqrt[21],​ k == 6 && 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}, {((13/60 + (13*I)/​60)*(4*Sqrt[5]*Ea2u1 - 4*Sqrt[5]*Ea2u2 + 9*Sqrt[5]*Eeu1 - 9*Sqrt[5]*Eeu2 + 2*Ma2u + 36*Meu))/​Sqrt[21],​ k == 6 && m == 3}, {((-13*I)/​60)*Sqrt[11/​21]*(9*Ea1u - 4*Ea2u1 - 5*Ea2u2 - 4*Sqrt[5]*Ma2u),​ k == 6 && m == 6}}, 0] + + ​ + + ### + + <​hidden **Input format suitable for 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, (-1/​2+-1/​2*I)*((1/​(sqrt(7)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + Ma2u + (-4)*(Meu)))} , + ​{4,​-3,​ (1/​2+-1/​2*I)*((1/​(sqrt(7)))*((2)*((sqrt(5))*(Ea2u1)) + (-2)*((sqrt(5))*(Ea2u2)) + (-1)*((sqrt(5))*(Eeu1)) + (sqrt(5))*(Eeu2) + 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/​60+13/​60*I)*((1/​(sqrt(21)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (2)*(Ma2u) + (36)*(Meu)))} , + {6, 3, (13/​60+13/​60*I)*((1/​(sqrt(21)))*((4)*((sqrt(5))*(Ea2u1)) + (-4)*((sqrt(5))*(Ea2u2)) + (9)*((sqrt(5))*(Eeu1)) + (-9)*((sqrt(5))*(Eeu2)) + (2)*(Ma2u) + (36)*(Meu)))} , + {6, 6, (-13/​60*I)*((sqrt(11/​21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (-4)*((sqrt(5))*(Ma2u))))} , + ​{6,​-6,​ (13/​60*I)*((sqrt(11/​21))*((9)*(Ea1u) + (-4)*(Ea2u1) + (-5)*(Ea2u2) + (-4)*((sqrt(5))*(Ma2u))))} } + + ​ + + ### + + ​ + + + ### + + |    ^  ${Y_{-3}^{(3)}}$  ^  ${Y_{-2}^{(3)}}$  ^  ${Y_{-1}^{(3)}}$  ^  ${Y_{0}^{(3)}}$  ^  ${Y_{1}^{(3)}}$  ^  ${Y_{2}^{(3)}}$  ^  ${Y_{3}^{(3)}}$  ^ + ^${Y_{-3}^{(3)}}$|$\frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right)$|$0$|$0$|$\left(-\frac{1}{18}+\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}\right)$|$0$|$0$|$\frac{1}{18} i \left(-9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right)$| + ^${Y_{-2}^{(3)}}$|$0$|$\frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}-2 \sqrt{5} \text{Meu}\right)$|$0$|$0$|$-\frac{1}{6} (-1)^{3/4} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$|$0$|$0$| + ^${Y_{-1}^{(3)}}$|$0$|$0$|$\frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}+2 \sqrt{5} \text{Meu}\right)$|$0$|$0$|$\frac{1}{6} (-1)^{3/4} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$|$0$| + ^${Y_{0}^{(3)}}$|$\left(-\frac{1}{18}-\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}\right)$|$0$|$0$|$\frac{1}{9} \left(5 \text{Ea2u1}+4 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right)$|$0$|$0$|$\left(\frac{1}{18}-\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}\right)$| + ^${Y_{1}^{(3)}}$|$0$|$\frac{1}{6} \sqrt[4]{-1} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$|$0$|$0$|$\frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}+2 \sqrt{5} \text{Meu}\right)$|$0$|$0$| + ^${Y_{2}^{(3)}}$|$0$|$0$|$-\frac{1}{6} \sqrt[4]{-1} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}+4 \text{Meu}\right)$|$0$|$0$|$\frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}-2 \sqrt{5} \text{Meu}\right)$|$0$| + ^${Y_{3}^{(3)}}$|$-\frac{1}{18} i \left(-9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right)$|$0$|$0$|$\left(\frac{1}{18}+\frac{i}{18}\right) \left(2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}\right)$|$0$|$0$|$\frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right)$| + + + ### + + ​ + + + ### + + |    ^  $f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$  ^  $f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$  ^  $f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$  ^  $f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^ + ^$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$|$\text{Ea1u}$|$0$|$0$|$0$|$0$|$0$|$0$| + ^$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$|$0$|$\text{Ea2u1}$|$\text{Ma2u}$|$0$|$0$|$0$|$0$| + ^$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$|$0$|$\text{Ma2u}$|$\text{Ea2u2}$|$0$|$0$|$0$|$0$| + ^$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$0$|$0$|$\text{Eeu1}$|$0$|$\text{Meu}$|$0$| + ^$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$0$|$0$|$0$|$\text{Eeu1}$|$0$|$\text{Meu}$| + ^$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$0$|$0$|$\text{Meu}$|$0$|$\text{Eeu2}$|$0$| + ^$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$0$|$0$|$0$|$\text{Meu}$|$0$|$\text{Eeu2}$| + + + ### + + ​ + + + ### + + |    ^  ${Y_{-3}^{(3)}}$  ^  ${Y_{-2}^{(3)}}$  ^  ${Y_{-1}^{(3)}}$  ^  ${Y_{0}^{(3)}}$  ^  ${Y_{1}^{(3)}}$  ^  ${Y_{2}^{(3)}}$  ^  ${Y_{3}^{(3)}}$  ^ + ^$f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$|$\frac{1}{2}+\frac{i}{2}$|$0$|$0$|$0$|$0$|$0$|$-\frac{1}{2}+\frac{i}{2}$| + ^$f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$|$\frac{1}{3}-\frac{i}{3}$|$0$|$0$|$-\frac{\sqrt{5}}{3}$|$0$|$0$|$-\frac{1}{3}-\frac{i}{3}$| + ^$f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$|$\left(\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$|$0$|$0$|$\frac{2}{3}$|$0$|$0$|$\left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt{5}$| + ^$f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$-\frac{1}{2 \sqrt{3}}$|$0$|$\frac{1}{2 \sqrt{3}}$|$\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$0$| + ^$f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$-\frac{i}{2 \sqrt{3}}$|$0$|$-\frac{i}{2 \sqrt{3}}$|$\left(-\frac{1}{2}+\frac{i}{2}\right) \sqrt{\frac{5}{6}}$|$0$| + ^$f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$-\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$|$-\frac{\sqrt{\frac{5}{3}}}{2}$|$0$|$\frac{\sqrt{\frac{5}{3}}}{2}$|$-\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$|$0$| + ^$f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$|$0$|$\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{6}}$|$-\frac{1}{2} i \sqrt{\frac{5}{3}}$|$0$|$-\frac{1}{2} i \sqrt{\frac{5}{3}}$|$\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{6}}$|$0$| + + + ### + + ​ + + + ### + + ^ ^$$\text{Ea1u}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_1.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \sin ^3(\theta ) (\sin (3 \phi )+\cos (3 \phi ))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^3+3 x^2 y-3 x y^2-y^3\right)$$ | ::: | + ^ ^$$\text{Ea2u1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_2.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{24}+\frac{i}{24}\right) \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \left(e^{6 i \phi } \sin ^3(\theta )-(1-i) e^{3 i \phi } \cos (\theta ) \left(5 \cos ^2(\theta )-3\right)-i \sin ^3(\theta )\right)$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(x^3-3 x^2 y-3 x y^2+y^3-5 z^3+3 z\right)$$ | ::: | + ^ ^$$\text{Ea2u2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_3.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\left(\frac{1}{48}+\frac{i}{48}\right) \sqrt{\frac{7}{\pi }} e^{-3 i \phi } \left(5 \left(e^{6 i \phi }-i\right) \sin ^3(\theta )+(2-2 i) e^{3 i \phi } \cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(5 x^3-15 x^2 y-15 x y^2+5 y^3+4 z \left(5 z^2-3\right)\right)$$ | ::: | + ^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_4.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )-\cos (2 \phi )))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(5 x^2 z-10 x y z-5 x z^2+x-5 y^2 z\right)$$ | ::: | + ^ ^$$\text{Eeu1}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_5.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \sin (\phi )+5 \sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi )))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 x^2 z-10 x y z+5 y^2 z-5 y z^2+y\right)$$ | ::: | + ^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_6.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) ((5 \cos (2 \theta )+3) \cos (\phi )+\sin (2 \theta ) (\cos (2 \phi )-\sin (2 \phi )))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 (-z)+2 x y z-5 x z^2+x+y^2 z\right)$$ | ::: | + ^ ^$$\text{Eeu2}$$ | {{:​physics_chemistry:​pointgroup:​d3d_z(x-y)_a_orb_3_7.png?​150}} | + |$$\psi(\theta,​\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) (\sin (2 \theta ) (\sin (2 \phi )+\cos (2 \phi ))-(5 \cos (2 \theta )+3) \sin (\phi ))$$ | ::: | + |$$\psi(\hat{x},​\hat{y},​\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(x^2 z+2 x y z-y^2 z-5 y z^2+y\right)$$ | ::: | + + + ### + + ​ + ===== Coupling between two shells ===== + + + + ### + + Click on one of the subsections to expand it or <​hiddenSwitch expand all> ​ + + ### + + ==== 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] + + ​ + + ### + + <​hidden **Input format suitable for Quanty** > + + ### + + ​ + + Akm = {{2, 0, (sqrt(5))*(Ma1g)} } + + ​ + + ### + + ​ + + + ### + + |    ^  ${Y_{-2}^{(2)}}$  ^  ${Y_{-1}^{(2)}}$  ^  ${Y_{0}^{(2)}}$  ^  ${Y_{1}^{(2)}}$  ^  ${Y_{2}^{(2)}}$  ^ + ^${Y_{0}^{(0)}}$|$0$|$0$|$\text{Ma1g}$|$0$|$0$| + + + ### + + ​ + + + ### + + |    ^  $d_{3z^2-r^2}$  ^  $d_{(x-y)(x+y+z)}$  ^  $d_{2\text{xy}-\text{xz}-\text{yz}}$  ^  $d_{(x-y)(x+y-2z)}$  ^  $d_{\text{yz}+\text{xz}+\text{xy}}$  ^ + ^$\text{s}$|$\text{Ma1g}$|$0$|$0$|$0$|$0$| + + + ### + + ​ + ==== Potential for p-f orbital mixing ==== + + + + ### + + ​$$A_{k,​m} = \begin{cases} + 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ + ​\frac{5 \left(\sqrt{5} \text{Ma2u1}+4 \text{Ma2u2}-4 \text{Meu1}\right)}{\sqrt{21}} & k=2\land m=0 \\ + ​\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) & k=4\land m=-3 \\ + ​-\frac{1}{2} \sqrt{\frac{3}{7}} \left(8 \sqrt{5} \text{Ma2u1}+11 \text{Ma2u2}-18 \text{Meu1}\right) & k=4\land m=0 \\ + ​\left(-\frac{1}{2}-\frac{i}{2}\right) \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) & \text{True} + \end{cases}$$ + + ### + + ​ + + + ### + + ​ + + 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}, {(1/2 - I/​2)*Sqrt[3]*(2*Ma2u1 + Sqrt[5]*Ma2u2),​ k == 4 && m == -3}, {-(Sqrt[3/​7]*(8*Sqrt[5]*Ma2u1 + 11*Ma2u2 - 18*Meu1))/​2,​ k == 4 && m == 0}}, (-1/2 - I/​2)*Sqrt[3]*(2*Ma2u1 + Sqrt[5]*Ma2u2)] + + ​ + + ### + + <​hidden **Input format suitable for 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+-1/​2*I)*((sqrt(3))*((2)*(Ma2u1) + (sqrt(5))*(Ma2u2)))} , + ​{4,​-3,​ (1/​2+-1/​2*I)*((sqrt(3))*((2)*(Ma2u1) + (sqrt(5))*(Ma2u2)))} } + + ​ + + ### + + ​ + + + ### + + |    ^  ${Y_{-3}^{(3)}}$  ^  ${Y_{-2}^{(3)}}$  ^  ${Y_{-1}^{(3)}}$  ^  ${Y_{0}^{(3)}}$  ^  ${Y_{1}^{(3)}}$  ^  ${Y_{2}^{(3)}}$  ^  ${Y_{3}^{(3)}}$  ^ + ^${Y_{-1}^{(1)}}$|$0$|$0$|$\frac{2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}}$|$0$|$0$|$\left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]$|$0$| + ^${Y_{0}^{(1)}}$|$\left(\frac{1}{6}+\frac{i}{6}\right) \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right)$|$0$|$0$|$\frac{1}{3} \left(2 \text{Ma2u2}-\sqrt{5} \text{Ma2u1}\right)$|$0$|$0$|$\left(-\frac{1}{6}+\frac{i}{6}\right) \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right)$| + ^${Y_{1}^{(1)}}$|$0$|$\left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]$|$0$|$0$|$\frac{2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}}{\sqrt{6}}$|$0$|$0$| + + + ### + + ​ + + + ### + + |    ^  $f_{(x-y)\left\backslash \left(x^2+4\backslash x\backslash y+y^2\right)\right.}$  ^  $f_{(x+y+z)\left\backslash \left(x^2-4\backslash x\backslash y+y^2+2\backslash (x+y)\backslash z-2\left\backslash z^2\right.\right)\right.}$  ^  $f_{\left(5\backslash (x+y)\left\backslash \left(x^2-4\backslash x\backslash y+y^2\right)\right.-\left.12\left\backslash \left(x^2+y^2\right)\right.\right\backslash z+8\left\backslash z^3\right.\right)}$  ^  $f_{\left(x^3+\left.5\left\backslash x^2\right.\right\backslash z-\left.5\left\backslash y^2\right.\right\backslash z+x\left\backslash \left(y^2-10\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(\left.x^2\right\backslash (y-5\backslash z)-10\backslash x\backslash y\backslash z+y\left\backslash \left(y^2+5\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(x^3-\left.x^2\right\backslash z+\left.y^2\right\backslash z+x\left\backslash \left(y^2+2\backslash y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^  $f_{\left(2\backslash x\backslash y\backslash z+\left.x^2\right\backslash (y+z)+y\left\backslash \left(y^2-y\backslash z-4\left\backslash z^2\right.\right)\right.\right)}$  ^ + ^$p_z$|$0$|$\text{Ma2u1}$|$\text{Ma2u2}$|$0$|$0$|$0$|$0$| + ^$p_x$|$0$|$0$|$0$|$\left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+i \sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right)$|$\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\frac{5}{3}} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)$|$\left(\frac{1}{12}+\frac{i}{12}\right) \left(-\sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-i \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \sqrt{5} \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right)$|$-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)}{\sqrt{3}}$| + ^$p_y$|$0$|$0$|$0$|$\left(-\frac{1}{4}-\frac{i}{4}\right) \sqrt{\frac{5}{3}} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)$|$\left(\frac{1}{12}+\frac{i}{12}\right) \left(\sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]+i \sqrt{15} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right)$|$\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \left(\text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]+i \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]\right)}{\sqrt{3}}$|$\left(\frac{1}{12}+\frac{i}{12}\right) \left(-\sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​3\right]-i \sqrt{3} \left(2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}\right) \text{Root}\left[36 \text{$\#​$1}^4+1\$|$,​1\right]-(1-i) \sqrt{5} \left(2 \sqrt{5} \text{Ma2u1}+5 \text{Ma2u2}-6 \text{Meu1}\right)\right)$| + + + ### + + ​ + + ===== Table of several point groups ===== + + ### + + [[physics_chemistry:​point_groups|Return to Main page on Point Groups]] + + ### + + ### + + ^Nonaxial groups ​     | [[physics_chemistry:​point_groups:​c1|C]]<​sub>​[[physics_chemistry:​point_groups:​c1|1]]​ | [[physics_chemistry:​point_groups:​cs|C]]<​sub>​[[physics_chemistry:​point_groups:​cs|s]]​ | [[physics_chemistry:​point_groups:​ci|C]]<​sub>​[[physics_chemistry:​point_groups:​ci|i]]​ | | | | | + ^C<​sub>​n​ groups | [[physics_chemistry:​point_groups:​c2|C]]<​sub>​[[physics_chemistry:​point_groups:​c2|2]]​ | [[physics_chemistry:​point_groups:​c3|C]]<​sub>​[[physics_chemistry:​point_groups:​c3|3]]​ | [[physics_chemistry:​point_groups:​c4|C]]<​sub>​[[physics_chemistry:​point_groups:​c4|4]]​ | [[physics_chemistry:​point_groups:​c5|C]]<​sub>​[[physics_chemistry:​point_groups:​c5|5]]​ | [[physics_chemistry:​point_groups:​c6|C]]<​sub>​[[physics_chemistry:​point_groups:​c6|6]]​ | [[physics_chemistry:​point_groups:​c7|C]]<​sub>​[[physics_chemistry:​point_groups:​c7|7]]​ | [[physics_chemistry:​point_groups:​c8|C]]<​sub>​[[physics_chemistry:​point_groups:​c8|8]]​ | + ^D<​sub>​n​ groups | [[physics_chemistry:​point_groups:​d2|D]]<​sub>​[[physics_chemistry:​point_groups:​d2|2]]​ | [[physics_chemistry:​point_groups:​d3|D]]<​sub>​[[physics_chemistry:​point_groups:​d3|3]]​ | [[physics_chemistry:​point_groups:​d4|D]]<​sub>​[[physics_chemistry:​point_groups:​d4|4]]​ | [[physics_chemistry:​point_groups:​d5|D]]<​sub>​[[physics_chemistry:​point_groups:​d5|5]]​ | [[physics_chemistry:​point_groups:​d6|D]]<​sub>​[[physics_chemistry:​point_groups:​d6|6]]​ | [[physics_chemistry:​point_groups:​d7|D]]<​sub>​[[physics_chemistry:​point_groups:​d7|7]]​ | [[physics_chemistry:​point_groups:​d8|D]]<​sub>​[[physics_chemistry:​point_groups:​d8|8]]​ | + ^C<​sub>​nv​ groups | [[physics_chemistry:​point_groups:​c2v|C]]<​sub>​[[physics_chemistry:​point_groups:​c2v|2v]]​ | [[physics_chemistry:​point_groups:​c3v|C]]<​sub>​[[physics_chemistry:​point_groups:​c3v|3v]]​ | [[physics_chemistry:​point_groups:​c4v|C]]<​sub>​[[physics_chemistry:​point_groups:​c4v|4v]]​ | [[physics_chemistry:​point_groups:​c5v|C]]<​sub>​[[physics_chemistry:​point_groups:​c5v|5v]]​ | [[physics_chemistry:​point_groups:​c6v|C]]<​sub>​[[physics_chemistry:​point_groups:​c6v|6v]]​ | [[physics_chemistry:​point_groups:​c7v|C]]<​sub>​[[physics_chemistry:​point_groups:​c7v|7v]]​ | [[physics_chemistry:​point_groups:​c8v|C]]<​sub>​[[physics_chemistry:​point_groups:​c8v|8v]]​ | + ^C<​sub>​nh​ groups | [[physics_chemistry:​point_groups:​c2h|C]]<​sub>​[[physics_chemistry:​point_groups:​c2h|2h]]​ | [[physics_chemistry:​point_groups:​c3h|C]]<​sub>​[[physics_chemistry:​point_groups:​c3h|3h]]​ | [[physics_chemistry:​point_groups:​c4h|C]]<​sub>​[[physics_chemistry:​point_groups:​c4h|4h]]​ | [[physics_chemistry:​point_groups:​c5h|C]]<​sub>​[[physics_chemistry:​point_groups:​c5h|5h]]​ | [[physics_chemistry:​point_groups:​c6h|C]]<​sub>​[[physics_chemistry:​point_groups:​c6h|6h]]​ | | | + ^D<​sub>​nh​ groups | [[physics_chemistry:​point_groups:​d2h|D]]<​sub>​[[physics_chemistry:​point_groups:​d2h|2h]]​ | [[physics_chemistry:​point_groups:​d3h|D]]<​sub>​[[physics_chemistry:​point_groups:​d3h|3h]]​ | [[physics_chemistry:​point_groups:​d4h|D]]<​sub>​[[physics_chemistry:​point_groups:​d4h|4h]]​ | [[physics_chemistry:​point_groups:​d5h|D]]<​sub>​[[physics_chemistry:​point_groups:​d5h|5h]]​ | [[physics_chemistry:​point_groups:​d6h|D]]<​sub>​[[physics_chemistry:​point_groups:​d6h|6h]]​ | [[physics_chemistry:​point_groups:​d7h|D]]<​sub>​[[physics_chemistry:​point_groups:​d7h|7h]]​ | [[physics_chemistry:​point_groups:​d8h|D]]<​sub>​[[physics_chemistry:​point_groups:​d8h|8h]]​ | + ^D<​sub>​nd​ groups | [[physics_chemistry:​point_groups:​d2d|D]]<​sub>​[[physics_chemistry:​point_groups:​d2d|2d]]​ | [[physics_chemistry:​point_groups:​d3d|D]]<​sub>​[[physics_chemistry:​point_groups:​d3d|3d]]​ | [[physics_chemistry:​point_groups:​d4d|D]]<​sub>​[[physics_chemistry:​point_groups:​d4d|4d]]​ | [[physics_chemistry:​point_groups:​d5d|D]]<​sub>​[[physics_chemistry:​point_groups:​d5d|5d]]​ | [[physics_chemistry:​point_groups:​d6d|D]]<​sub>​[[physics_chemistry:​point_groups:​d6d|6d]]​ | [[physics_chemistry:​point_groups:​d7d|D]]<​sub>​[[physics_chemistry:​point_groups:​d7d|7d]]​ | [[physics_chemistry:​point_groups:​d8d|D]]<​sub>​[[physics_chemistry:​point_groups:​d8d|8d]]​ | + ^S<​sub>​n​ groups | [[physics_chemistry:​point_groups:​S2|S]]<​sub>​[[physics_chemistry:​point_groups:​S2|2]]​ | [[physics_chemistry:​point_groups:​S4|S]]<​sub>​[[physics_chemistry:​point_groups:​S4|4]]​ | [[physics_chemistry:​point_groups:​S6|S]]<​sub>​[[physics_chemistry:​point_groups:​S6|6]]​ | [[physics_chemistry:​point_groups:​S8|S]]<​sub>​[[physics_chemistry:​point_groups:​S8|8]]​ | [[physics_chemistry:​point_groups:​S10|S]]<​sub>​[[physics_chemistry:​point_groups:​S10|10]]​ | [[physics_chemistry:​point_groups:​S12|S]]<​sub>​[[physics_chemistry:​point_groups:​S12|12]]​ |  | + ^Cubic groups | [[physics_chemistry:​point_groups:​T|T]] | [[physics_chemistry:​point_groups:​Th|T]]<​sub>​[[physics_chemistry:​point_groups:​Th|h]]​ | [[physics_chemistry:​point_groups:​Td|T]]<​sub>​[[physics_chemistry:​point_groups:​Td|d]]​ | [[physics_chemistry:​point_groups:​O|O]] | [[physics_chemistry:​point_groups:​Oh|O]]<​sub>​[[physics_chemistry:​point_groups:​Oh|h]]​ | [[physics_chemistry:​point_groups:​I|I]] | [[physics_chemistry:​point_groups:​Ih|I]]<​sub>​[[physics_chemistry:​point_groups:​Ih|h]]​ | + ^Linear groups ​     | [[physics_chemistry:​point_groups:​cinfv|C]]<​sub>​[[physics_chemistry:​point_groups:​cinfv|$\infty$v]]​ | [[physics_chemistry:​point_groups:​cinfv|D]]<​sub>​[[physics_chemistry:​point_groups:​dinfh|$\infty$h]]​ | | | | | | + + ###