~~CLOSETOC~~ ====== Orientation 0sqrt2-1z ====== ===== Symmetry Operations ===== ### In the Oh Point Group, with orientation 0sqrt2-1z there are the following symmetry operations ### ### {{:physics_chemistry:pointgroup:oh_0sqrt2-1z.png}} ### ### ^ Operator ^ Orientation ^ ^ $\text{E}$ | $\{0,0,0\}$ , | ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\sqrt{6},-\sqrt{2},1\right\}$ , $\left\{0,-2 \sqrt{2},-1\right\}$ , $\left\{-\sqrt{6},\sqrt{2},-1\right\}$ , $\left\{\sqrt{6},\sqrt{2},-1\right\}$ , $\left\{0,2 \sqrt{2},1\right\}$ , $\left\{\sqrt{6},-\sqrt{2},1\right\}$ , | ^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ , $\left\{0,1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},1,-2 \sqrt{2}\right\}$ , $\left\{-\sqrt{3},1,-2 \sqrt{2}\right\}$ , | ^ $C_4$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{0,\sqrt{2},-1\right\}$ , $\left\{-\sqrt{3},1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},-1,-\sqrt{2}\right\}$ , $\left\{-\sqrt{3},-1,-\sqrt{2}\right\}$ , | ^ $C_2$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{-\sqrt{3},1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},1,\sqrt{2}\right\}$ , | ^ $\text{i}$ | $\{0,0,0\}$ , | ^ $S_4$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{0,\sqrt{2},-1\right\}$ , $\left\{-\sqrt{3},1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},-1,-\sqrt{2}\right\}$ , $\left\{-\sqrt{3},-1,-\sqrt{2}\right\}$ , | ^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\sqrt{6},-\sqrt{2},1\right\}$ , $\left\{0,-2 \sqrt{2},-1\right\}$ , $\left\{-\sqrt{6},\sqrt{2},-1\right\}$ , $\left\{\sqrt{6},\sqrt{2},-1\right\}$ , $\left\{0,2 \sqrt{2},1\right\}$ , $\left\{\sqrt{6},-\sqrt{2},1\right\}$ , | ^ $\sigma _h$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{-\sqrt{3},1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},1,\sqrt{2}\right\}$ , | ^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ , $\left\{0,1,\sqrt{2}\right\}$ , $\left\{\sqrt{3},1,-2 \sqrt{2}\right\}$ , $\left\{-\sqrt{3},1,-2 \sqrt{2}\right\}$ , | ### ===== Different Settings ===== ### * [[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]] * [[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]] * [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]] * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]] * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] ### ===== Character Table ===== ### | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_3 \,{\text{(8)}} $ ^ $ C_2 \,{\text{(6)}} $ ^ $ C_4 \,{\text{(6)}} $ ^ $ C_2 \,{\text{(3)}} $ ^ $ \text{i} \,{\text{(1)}} $ ^ $ S_4 \,{\text{(6)}} $ ^ $ S_6 \,{\text{(8)}} $ ^ $ \sigma_h \,{\text{(3)}} $ ^ $ \sigma_d \,{\text{(6)}} $ ^ ^ $ A_{1g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | ^ $ A_{2g} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | ^ $ E_g $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 0 $ | $ 2 $ | $ 2 $ | $ 0 $ | $ -1 $ | $ 2 $ | $ 0 $ | ^ $ T_{1g} $ | $ 3 $ | $ 0 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 3 $ | $ 1 $ | $ 0 $ | $ -1 $ | $ -1 $ | ^ $ T_{2g} $ | $ 3 $ | $ 0 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 3 $ | $ -1 $ | $ 0 $ | $ -1 $ | $ 1 $ | ^ $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | ^ $ A_{2u} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | ^ $ E_u $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 0 $ | $ 2 $ | $ -2 $ | $ 0 $ | $ 1 $ | $ -2 $ | $ 0 $ | ^ $ T_{1u} $ | $ 3 $ | $ 0 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ -3 $ | $ -1 $ | $ 0 $ | $ 1 $ | $ 1 $ | ^ $ T_{2u} $ | $ 3 $ | $ 0 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -3 $ | $ 1 $ | $ 0 $ | $ 1 $ | $ -1 $ | ### ===== Product Table ===== ### | $ $ ^ $ A_{1g} $ ^ $ A_{2g} $ ^ $ E_g $ ^ $ T_{1g} $ ^ $ T_{2g} $ ^ $ A_{1u} $ ^ $ A_{2u} $ ^ $ E_u $ ^ $ T_{1u} $ ^ $ T_{2u} $ ^ ^ $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ T_{1g} $ | $ T_{2g} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ T_{1u} $ | $ T_{2u} $ | ^ $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ T_{2g} $ | $ T_{1g} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ T_{2u} $ | $ T_{1u} $ | ^ $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ T_{1g}+T_{2g} $ | $ T_{1g}+T_{2g} $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ T_{1u}+T_{2u} $ | $ T_{1u}+T_{2u} $ | ^ $ T_{1g} $ | $ T_{1g} $ | $ T_{2g} $ | $ T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | $ T_{1u} $ | $ T_{2u} $ | $ T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | ^ $ T_{2g} $ | $ T_{2g} $ | $ T_{1g} $ | $ T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | $ T_{2u} $ | $ T_{1u} $ | $ T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | ^ $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ T_{1u} $ | $ T_{2u} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ T_{1g} $ | $ T_{2g} $ | ^ $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ T_{2u} $ | $ T_{1u} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ T_{2g} $ | $ T_{1g} $ | ^ $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ T_{1u}+T_{2u} $ | $ T_{1u}+T_{2u} $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ T_{1g}+T_{2g} $ | $ T_{1g}+T_{2g} $ | ^ $ T_{1u} $ | $ T_{1u} $ | $ T_{2u} $ | $ T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | $ T_{1g} $ | $ T_{2g} $ | $ T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | ^ $ T_{2u} $ | $ T_{2u} $ | $ T_{1u} $ | $ T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | $ T_{2g} $ | $ T_{1g} $ | $ T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | ### ===== Sub Groups with compatible settings ===== ### * [[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:d3d:orientation_zx_b|Point Group D3d with orientation Zx_B]] * [[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]] ### ===== Super Groups with compatible settings ===== ### ### ===== Invariant Potential expanded on renormalized spherical Harmonics ===== ### Any potential (function) can be written as a sum over spherical harmonics. $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Oh Point group with orientation 0sqrt2-1z the form of the expansion coefficients is: ### ==== Expansion ==== ### $$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ -i \sqrt{\frac{10}{7}} A(4,0) & k=4\land (m=-3\lor m=3) \\ A(4,0) & k=4\land m=0 \\ -\frac{1}{8} \sqrt{\frac{77}{3}} A(6,0) & k=6\land (m=-6\lor m=6) \\ \frac{1}{4} i \sqrt{\frac{35}{6}} A(6,0) & k=6\land (m=-3\lor m=3) \\ A(6,0) & k=6\land m=0 \end{cases}$$ ### ==== Input format suitable for Mathematica (Quanty.nb) ==== ### Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {(-I)*Sqrt[10/7]*A[4, 0], k == 4 && (m == -3 || m == 3)}, {A[4, 0], k == 4 && m == 0}, {-(Sqrt[77/3]*A[6, 0])/8, k == 6 && (m == -6 || m == 6)}, {(I/4)*Sqrt[35/6]*A[6, 0], k == 6 && (m == -3 || m == 3)}, {A[6, 0], k == 6 && m == 0}}, 0] ### ==== Input format suitable for Quanty ==== ### Akm = {{0, 0, A(0,0)} , {4, 0, A(4,0)} , {4,-3, (-I)*((sqrt(10/7))*(A(4,0)))} , {4, 3, (-I)*((sqrt(10/7))*(A(4,0)))} , {6, 0, A(6,0)} , {6,-3, (1/4*I)*((sqrt(35/6))*(A(6,0)))} , {6, 3, (1/4*I)*((sqrt(35/6))*(A(6,0)))} , {6,-6, (-1/8)*((sqrt(77/3))*(A(6,0)))} , {6, 6, (-1/8)*((sqrt(77/3))*(A(6,0)))} } ### ==== One particle coupling on a basis of spherical harmonics ==== ### The operator representing the potential in second quantisation is given as: $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$ ### ### we can express the operator as $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ ### ### The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter. ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ -\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0) $| ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $| ^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \frac{5}{21} i \sqrt{2} \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_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ -\frac{5}{21} i \sqrt{2} \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_{0}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{5}{21} i \sqrt{2} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{5}{21} i \sqrt{2} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{1}{11} i \sqrt{10} \text{Aff}(4,0)+\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{35}{156} \text{Aff}(6,0) $| ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{2}{33} i \sqrt{5} \text{Aff}(4,0)-\frac{35}{572} i \sqrt{5} \text{Aff}(6,0) $|$ 0 $|$ 0 $| ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{35}{572} i \sqrt{5} \text{Aff}(6,0)-\frac{2}{33} i \sqrt{5} \text{Aff}(4,0) $|$ 0 $| ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{11} i \sqrt{10} \text{Aff}(4,0)-\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{1}{11} i \sqrt{10} \text{Aff}(4,0)-\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0) $| ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{35}{572} i \sqrt{5} \text{Aff}(6,0)-\frac{2}{33} i \sqrt{5} \text{Aff}(4,0) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{2}{33} i \sqrt{5} \text{Aff}(4,0)-\frac{35}{572} i \sqrt{5} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{35}{156} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{1}{11} i \sqrt{10} \text{Aff}(4,0)+\frac{35}{858} i \sqrt{\frac{5}{2}} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| ### ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== ### Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{\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 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ p_x $|$\color{darkred}{ 0 }$|$ \text{App}(0,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{2 \text{Apf}(4,0)}{\sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{2 \text{Apf}(4,0)}{\sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \frac{2 \text{Apf}(4,0)}{\sqrt{21}} $|$ 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{3}{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^2+y^2-2\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)-\frac{3}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{\text{xz}-\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{-x^2+y^2+\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\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 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)+\frac{6}{11} \text{Aff}(4,0)+\frac{45}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 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{2 \text{Apf}(4,0)}{\sqrt{21}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 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{2 \text{Apf}(4,0)}{\sqrt{21}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 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 \text{Apf}(4,0)}{\sqrt{21}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{3}{11} \text{Aff}(4,0)+\frac{75}{572} \text{Aff}(6,0) $|$ 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 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0) $|$ 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 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0) $|$ 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}{11} \text{Aff}(4,0)-\frac{135}{572} \text{Aff}(6,0) $| ### ===== Coupling for a single shell ===== ### Although the parameters $A_{l'',l'}(k,m)$ uniquely define the potential, there is no simple relation between these paramters and the eigenstates of the potential. In this section we replace the parameters $A_{l'',l'}(k,m)$ by paramters that relate to the eigen energies of the potential acting on or between two shells with angular momentum $l''$ and $l'$. ### ### Click on one of the subsections to expand it or ### ==== Potential for s orbitals ==== ### $$A_{k,m} = \begin{cases} \text{Ea1g} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$$ ### ### 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:oh_0sqrt2-1z_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 ==== ### $$A_{k,m} = \begin{cases} \text{Et1u} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{Et1u, k == 0 && m == 0}}, 0] ### ### Akm = {{0, 0, Et1u} } ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ \text{Et1u} $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Et1u} $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Et1u} $| ### ### | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ ^$ p_x $|$ \text{Et1u} $|$ 0 $|$ 0 $| ^$ p_y $|$ 0 $|$ \text{Et1u} $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ \text{Et1u} $| ### ### | $ $ ^ $ {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 $| ### ### ^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_1_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | ^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_1_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | ^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{5} (2 \text{Eeg}+3 \text{Et2g}) & k=0\land m=0 \\ i \sqrt{\frac{14}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land (m=-3\lor m=3) \\ -\frac{7}{5} (\text{Eeg}-\text{Et2g}) & k=4\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(2*Eeg + 3*Et2g)/5, k == 0 && m == 0}, {I*Sqrt[14/5]*(Eeg - Et2g), k == 4 && (m == -3 || m == 3)}, {(-7*(Eeg - Et2g))/5, k == 4 && m == 0}}, 0] ### ### Akm = {{0, 0, (1/5)*((2)*(Eeg) + (3)*(Et2g))} , {4, 0, (-7/5)*(Eeg + (-1)*(Et2g))} , {4,-3, (I)*((sqrt(14/5))*(Eeg + (-1)*(Et2g)))} , {4, 3, (I)*((sqrt(14/5))*(Eeg + (-1)*(Et2g)))} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-2}^{(2)}} $|$ \frac{1}{3} (\text{Eeg}+2 \text{Et2g}) $|$ 0 $|$ 0 $|$ -\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ \frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $| ^$ {Y_{2}^{(2)}} $|$ 0 $|$ -\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (\text{Eeg}+2 \text{Et2g}) $| ### ### | $ $ ^ $ 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} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{-x^2+y^2-2\sqrt{2}\text{yz}} $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{\text{xz}-\sqrt{2}\text{xy}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $| ^$ d_{-x^2+y^2+\sqrt{2}\text{yz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $| ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $| ### ### | $ $ ^ $ {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 $| ### ### ^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sin (\theta ) \sin (\phi )+\sqrt{2} \cos (\theta )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(y+\sqrt{2} z\right)$$ | ::: | ^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sin (\theta ) \cos (2 \phi )+2 \sqrt{2} \cos (\theta ) \sin (\phi )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(x^2-y \left(y-2 \sqrt{2} z\right)\right)$$ | ::: | ^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} \sin (\theta ) \cos (\phi ) \left(\sqrt{2} \sin (\theta ) \sin (\phi )-\cos (\theta )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{5}{\pi }} x \left(\sqrt{2} y-z\right)$$ | ::: | ^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_2_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{5}{\pi }} \sin (\theta ) \left(\sqrt{2} \sin (\theta ) \cos (2 \phi )-2 \cos (\theta ) \sin (\phi )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(y \left(\sqrt{2} y+2 z\right)-\sqrt{2} x^2\right)$$ | ::: | ^ ^$$\text{Et2g}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea2u}+3 (\text{Et1u}+\text{Et2u})) & k=0\land m=0 \\ -i \sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land (m=-3\lor m=3) \\ \frac{1}{2} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land m=0 \\ -\frac{13}{60} \sqrt{\frac{11}{21}} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land (m=-6\lor m=6) \\ \frac{13 i (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u})}{6 \sqrt{210}} & k=6\land (m=-3\lor m=3) \\ \frac{26}{105} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) & k=6\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea2u + 3*(Et1u + Et2u))/7, k == 0 && m == 0}, {(-I)*Sqrt[5/14]*(2*Ea2u - 3*Et1u + Et2u), k == 4 && (m == -3 || m == 3)}, {(2*Ea2u - 3*Et1u + Et2u)/2, k == 4 && m == 0}, {(-13*Sqrt[11/21]*(4*Ea2u + 5*Et1u - 9*Et2u))/60, k == 6 && (m == -6 || m == 6)}, {(((13*I)/6)*(4*Ea2u + 5*Et1u - 9*Et2u))/Sqrt[210], k == 6 && (m == -3 || m == 3)}, {(26*(4*Ea2u + 5*Et1u - 9*Et2u))/105, k == 6 && m == 0}}, 0] ### ### Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} , {4, 0, (1/2)*((2)*(Ea2u) + (-3)*(Et1u) + Et2u)} , {4,-3, (-I)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , {4, 3, (-I)*((sqrt(5/14))*((2)*(Ea2u) + (-3)*(Et1u) + Et2u))} , {6, 0, (26/105)*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u))} , {6,-3, (13/6*I)*((1/(sqrt(210)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , {6, 3, (13/6*I)*((1/(sqrt(210)))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , {6,-6, (-13/60)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} , {6, 6, (-13/60)*((sqrt(11/21))*((4)*(Ea2u) + (5)*(Et1u) + (-9)*(Et2u)))} } ### ### | $ $ ^ $ {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} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) $| ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $|$ 0 $|$ -\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $| ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $| ^$ {Y_{0}^{(3)}} $|$ -\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} (5 \text{Ea2u}+4 \text{Et1u}) $|$ 0 $|$ 0 $|$ -\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $| ^$ {Y_{1}^{(3)}} $|$ 0 $|$ \frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ -\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u}) $| ### ### | $ $ ^ $ 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{Ea2u} $|$ 0 $|$ 0 $|$ 0 $|$ 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{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 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{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 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.} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 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 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 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 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $| ^$ f_{-x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $| ### ### | $ $ ^ $ {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}} $| ### ### ^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{35}{\pi }} \left(2 \sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+\cos (\theta ) (5 \cos (2 \theta )-1)\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{12} \sqrt{\frac{35}{\pi }} \left(3 \sqrt{2} x^2 y-\sqrt{2} y^3+5 z^3-3 z\right)$$ | ::: | ^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{7}{\pi }} x \left(10 \sqrt{2} y z+5 z^2-1\right)$$ | ::: | ^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \left(5 \sqrt{2} \sin (2 \theta ) \cos (2 \phi )+(5 \cos (2 \theta )+3) \sin (\phi )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} x^2 z+5 \sqrt{2} y^2 z-5 y z^2+y\right)$$ | ::: | ^ ^$$\text{Et1u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-5 \sqrt{2} \sin ^3(\theta ) \sin (3 \phi )+3 \cos (\theta )+5 \cos (3 \theta )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{24} \sqrt{\frac{7}{\pi }} \left(-15 \sqrt{2} x^2 y+5 \sqrt{2} y^3+4 z \left(5 z^2-3\right)\right)$$ | ::: | ^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \cos (\phi ) \left(-2 \sqrt{2} \sin (2 \theta ) \sin (\phi )+5 \cos (2 \theta )+3\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{35}{\pi }} x \left(2 \sqrt{2} y z-5 z^2+1\right)$$ | ::: | ^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_orb_3_6.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{35}{\pi }} \sin (\theta ) \left((5 \cos (2 \theta )+3) \sin (\phi )-\sqrt{2} \sin (2 \theta ) \cos (2 \phi )\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{8} \sqrt{\frac{35}{\pi }} \left(\sqrt{2} x^2 z-\sqrt{2} y^2 z-5 y z^2+y\right)$$ | ::: | ^ ^$$\text{Et2u}$$ | {{:physics_chemistry:pointgroup:oh_0sqrt2-1z_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 p-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 4\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ -i \sqrt{\frac{15}{2}} \text{Mt1u} & k=4\land (m=-3\lor m=3) \\ \frac{\sqrt{21} \text{Mt1u}}{2} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 4 || (m != -3 && m != 0 && m != 3)}, {(-I)*Sqrt[15/2]*Mt1u, k == 4 && (m == -3 || m == 3)}}, (Sqrt[21]*Mt1u)/2] ### ### Akm = {{4, 0, (1/2)*((sqrt(21))*(Mt1u))} , {4,-3, (-I)*((sqrt(15/2))*(Mt1u))} , {4, 3, (-I)*((sqrt(15/2))*(Mt1u))} } ### ### | $ $ ^ $ {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{\text{Mt1u}}{\sqrt{6}} $|$ 0 $|$ 0 $|$ -i \sqrt{\frac{5}{6}} \text{Mt1u} $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ \frac{1}{3} i \sqrt{\frac{5}{2}} \text{Mt1u} $|$ 0 $|$ 0 $|$ \frac{2 \text{Mt1u}}{3} $|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{5}{2}} \text{Mt1u} $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ -i \sqrt{\frac{5}{6}} \text{Mt1u} $|$ 0 $|$ 0 $|$ -\frac{\text{Mt1u}}{\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{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_y $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $| ### ===== Table of several point groups ===== ### [[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]] | | | | | | ###