~~CLOSETOC~~ ====== Orientation Zxy ====== ===== Symmetry Operations ===== ### In the D4h Point Group, with orientation Zxy there are the following symmetry operations ### ### {{:physics_chemistry:pointgroup:d4h_zxy.png}} ### ### ^ Operator ^ Orientation ^ ^ $\text{E}$ | $\{0,0,0\}$ , | ^ $C_4$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | ^ $C_2$ | $\{0,0,1\}$ , | ^ $C_2$ | $\{0,1,0\}$ , $\{1,0,0\}$ , | ^ $C_2$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , | ^ $\text{i}$ | $\{0,0,0\}$ , | ^ $S_4$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | ^ $\sigma _h$ | $\{0,0,1\}$ , | ^ $\sigma _v$ | $\{1,0,0\}$ , $\{0,1,0\}$ , | ^ $\sigma _d$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , | ### ===== Different Settings ===== ### * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] ### ===== Character Table ===== ### | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_4 \,{\text{(2)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ $ C_2 \,{\text{(2)}} $ ^ $ C_2 \,{\text{(2)}} $ ^ $ \text{i} \,{\text{(1)}} $ ^ $ S_4 \,{\text{(2)}} $ ^ $ \sigma_h \,{\text{(1)}} $ ^ $ \sigma_v \,{\text{(2)}} $ ^ $ \sigma_d \,{\text{(2)}} $ ^ ^ $ A_{1g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | ^ $ A_{2g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | ^ $ B_{1g} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | ^ $ B_{2g} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | ^ $ E_g $ | $ 2 $ | $ 0 $ | $ -2 $ | $ 0 $ | $ 0 $ | $ 2 $ | $ 0 $ | $ -2 $ | $ 0 $ | $ 0 $ | ^ $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | ^ $ A_{2u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | ^ $ B_{1u} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | ^ $ B_{2u} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | ^ $ E_u $ | $ 2 $ | $ 0 $ | $ -2 $ | $ 0 $ | $ 0 $ | $ -2 $ | $ 0 $ | $ 2 $ | $ 0 $ | $ 0 $ | ### ===== Product Table ===== ### | $ $ ^ $ A_{1g} $ ^ $ A_{2g} $ ^ $ B_{1g} $ ^ $ B_{2g} $ ^ $ E_g $ ^ $ A_{1u} $ ^ $ A_{2u} $ ^ $ B_{1u} $ ^ $ B_{2u} $ ^ $ E_u $ ^ ^ $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ B_{1g} $ | $ B_{2g} $ | $ E_g $ | $ A_{1u} $ | $ A_{2u} $ | $ B_{1u} $ | $ B_{2u} $ | $ E_u $ | ^ $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ B_{2g} $ | $ B_{1g} $ | $ E_g $ | $ A_{2u} $ | $ A_{1u} $ | $ B_{2u} $ | $ B_{1u} $ | $ E_u $ | ^ $ B_{1g} $ | $ B_{1g} $ | $ B_{2g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ B_{1u} $ | $ B_{2u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | ^ $ B_{2g} $ | $ B_{2g} $ | $ B_{1g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ B_{2u} $ | $ B_{1u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | ^ $ E_g $ | $ E_g $ | $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+B_{1g}+B_{2g} $ | $ E_u $ | $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+B_{1u}+B_{2u} $ | ^ $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ B_{1u} $ | $ B_{2u} $ | $ E_u $ | $ A_{1g} $ | $ A_{2g} $ | $ B_{1g} $ | $ B_{2g} $ | $ E_g $ | ^ $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ B_{2u} $ | $ B_{1u} $ | $ E_u $ | $ A_{2g} $ | $ A_{1g} $ | $ B_{2g} $ | $ B_{1g} $ | $ E_g $ | ^ $ B_{1u} $ | $ B_{1u} $ | $ B_{2u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ B_{1g} $ | $ B_{2g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | ^ $ B_{2u} $ | $ B_{2u} $ | $ B_{1u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ B_{2g} $ | $ B_{1g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | ^ $ E_u $ | $ E_u $ | $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+B_{1u}+B_{2u} $ | $ E_g $ | $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+B_{1g}+B_{2g} $ | ### ===== Sub Groups with compatible settings ===== ### * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] * [[physics_chemistry:point_groups:c2h:orientation_z|Point Group C2h with orientation Z]] * [[physics_chemistry:point_groups:c2v:orientation_zxy|Point Group C2v with orientation Zxy]] * [[physics_chemistry:point_groups:c2:orientation_x|Point Group C2 with orientation X]] * [[physics_chemistry:point_groups:c2:orientation_y|Point Group C2 with orientation Y]] * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]] * [[physics_chemistry:point_groups:c4h:orientation_z|Point Group C4h with orientation Z]] * [[physics_chemistry:point_groups:c4v:orientation_zxy|Point Group C4v with orientation Zxy]] * [[physics_chemistry:point_groups:c4:orientation_z|Point Group C4 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:cs:orientation_y|Point Group Cs with orientation Y]] * [[physics_chemistry:point_groups:cs:orientation_z|Point Group Cs with orientation Z]] * [[physics_chemistry:point_groups:d2d:orientation_zxy|Point Group D2d with orientation Zxy]] * [[physics_chemistry:point_groups:d2h:orientation_xyz|Point Group D2h with orientation XYZ]] * [[physics_chemistry:point_groups:d2:orientation_xyz|Point Group D2 with orientation XYZ]] * [[physics_chemistry:point_groups:d4:orientation_zxy|Point Group D4 with orientation Zxy]] * [[physics_chemistry:point_groups:s4:orientation_z|Point Group S4 with orientation Z]] ### ===== Super Groups with compatible settings ===== ### * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] ### ===== Invariant Potential expanded on renormalized spherical Harmonics ===== ### Any potential (function) can be written as a sum over spherical harmonics. $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the D4h Point group with orientation Zxy the form of the expansion coefficients is: ### ==== Expansion ==== ### $$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(2,0) & k=2\land m=0 \\ A(4,4) & k=4\land (m=-4\lor m=4) \\ A(4,0) & k=4\land m=0 \\ A(6,4) & k=6\land (m=-4\lor m=4) \\ A(6,0) & k=6\land m=0 \end{cases}$$ ### ==== Input format suitable for Mathematica (Quanty.nb) ==== ### Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}, {A[6, 4], k == 6 && (m == -4 || m == 4)}, {A[6, 0], k == 6 && m == 0}}, 0] ### ==== Input format suitable for Quanty ==== ### Akm = {{0, 0, A(0,0)} , {2, 0, A(2,0)} , {4, 0, A(4,0)} , {4,-4, A(4,4)} , {4, 4, A(4,4)} , {6, 0, A(6,0)} , {6,-4, A(6,4)} , {6, 4, A(6,4)} } ### ==== 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 $|$ 0 $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $| ^$ {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 }$|$ 0 $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ 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 $| ^$ {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 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\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 $|$ 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)}} $|$ \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 }$|$ 0 $|$ 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 }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ 0 $|$ 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 $|$ 0 $|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$\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 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ 0 $| ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 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 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $| ^$ {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 }$|$ 0 $|$ 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 $|$ 0 $| ^$ {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 }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 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 $| ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{2 \text{Apf}(4,4)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Aff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| ### ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== ### Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| ### ==== One particle coupling on a basis of symmetry adapted functions ==== ### After rotation we find ### ### | $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ ^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 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)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$ 0 $| ^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $| ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)+\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$ 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_{3z^2-r^2} $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0,0)+\frac{2}{7} \text{Add}(2,0)+\frac{2}{7} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \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_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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 }$| ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Add}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)-\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)-\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4) $|$ 0 $|$ 0 $| ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{3}{10} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{\text{Apf}(4,0)}{2 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} \text{Apf}(4,4) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{2}{15} \text{Aff}(2,0)+\frac{3}{44} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{125 \text{Aff}(6,0)}{1716}-\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4) $|$ 0 $| ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \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{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ -\frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}+\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)-\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Aff}(2,0)}{\sqrt{15}}-\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)+\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)-\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)-\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4) $|$ 0 $|$ 0 $| ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3 \text{Apf}(2,0)}{2 \sqrt{35}}-\frac{1}{6} \sqrt{\frac{5}{7}} \text{Apf}(4,0)+\frac{\text{Apf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{\text{Aff}(2,0)}{\sqrt{15}}+\frac{1}{44} \sqrt{\frac{5}{3}} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{7}{6}} \text{Aff}(4,4)+\frac{35}{572} \sqrt{\frac{5}{3}} \text{Aff}(6,0)+\frac{5}{286} \sqrt{\frac{35}{6}} \text{Aff}(6,4) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{7}{132} \text{Aff}(4,0)-\frac{1}{22} \sqrt{\frac{35}{2}} \text{Aff}(4,4)-\frac{5}{44} \text{Aff}(6,0)+\frac{25}{286} \sqrt{\frac{7}{2}} \text{Aff}(6,4) $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{1}{33} \sqrt{70} \text{Aff}(4,4)+\frac{10}{143} \text{Aff}(6,0)+\frac{10}{143} \sqrt{14} \text{Aff}(6,4) $| ### ===== 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:d4h_zxy_orb_0_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: | ### ==== Potential for p orbitals ==== ### $$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea2u}+2 \text{Eeu}) & k=0\land m=0 \\ \frac{5 (\text{Ea2u}-\text{Eeu})}{3} & k=2\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea2u + 2*Eeu)/3, k == 0 && m == 0}, {(5*(Ea2u - Eeu))/3, k == 2 && m == 0}}, 0] ### ### Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , {2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} } ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ \text{Eeu} $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea2u} $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Eeu} $| ### ### | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ ^$ p_x $|$ \text{Eeu} $|$ 0 $|$ 0 $| ^$ p_y $|$ 0 $|$ \text{Eeu} $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ \text{Ea2u} $| ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| ### ### ^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_1_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | ^ ^$$\text{Eeu}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_1_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | ^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_1_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | ### ==== Potential for d orbitals ==== ### $$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea1g}+\text{Eb1g}+\text{Eb2g}+2 \text{Eeg}) & k=0\land m=0 \\ \text{Ea1g}-\text{Eb1g}-\text{Eb2g}+\text{Eeg} & k=2\land m=0 \\ \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eb1g}-\text{Eb2g}) & k=4\land (m=-4\lor m=4) \\ \frac{3}{10} (6 \text{Ea1g}+\text{Eb1g}+\text{Eb2g}-8 \text{Eeg}) & k=4\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea1g + Eb1g + Eb2g + 2*Eeg)/5, k == 0 && m == 0}, {Ea1g - Eb1g - Eb2g + Eeg, k == 2 && m == 0}, {(3*Sqrt[7/10]*(Eb1g - Eb2g))/2, k == 4 && (m == -4 || m == 4)}, {(3*(6*Ea1g + Eb1g + Eb2g - 8*Eeg))/10, k == 4 && m == 0}}, 0] ### ### Akm = {{0, 0, (1/5)*(Ea1g + Eb1g + Eb2g + (2)*(Eeg))} , {2, 0, Ea1g + (-1)*(Eb1g) + (-1)*(Eb2g) + Eeg} , {4, 0, (3/10)*((6)*(Ea1g) + Eb1g + Eb2g + (-8)*(Eeg))} , {4,-4, (3/2)*((sqrt(7/10))*(Eb1g + (-1)*(Eb2g)))} , {4, 4, (3/2)*((sqrt(7/10))*(Eb1g + (-1)*(Eb2g)))} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Eb1g}+\text{Eb2g}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eb1g}-\text{Eb2g}}{2} $| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea1g} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eeg} $|$ 0 $| ^$ {Y_{2}^{(2)}} $|$ \frac{\text{Eb1g}-\text{Eb2g}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eb1g}+\text{Eb2g}}{2} $| ### ### | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ ^$ d_{x^2-y^2} $|$ \text{Eb1g} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{3z^2-r^2} $|$ 0 $|$ \text{Ea1g} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eeg} $|$ 0 $| ^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb2g} $| ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| ### ### ^ ^$$\text{Eb1g}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | ^ ^$$\text{Ea1g}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | ^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | ^ ^$$\text{Eeg}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | ^ ^$$\text{Eb2g}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_2_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: | ### ==== Potential for f orbitals ==== ### $$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea2u}+\text{Eb1u}+\text{Eb2u}+2 \text{Eeu1}+2 \text{Eeu2}) & k=0\land m=0 \\ \frac{5}{7} \left(\text{Ea2u}-\text{Eeu1}+\sqrt{15} \text{Meu}\right) & k=2\land m=0 \\ -\frac{3}{56} \left(2 \sqrt{70} \text{Eb1u}-2 \sqrt{70} \text{Eb2u}-3 \sqrt{70} \text{Eeu1}+3 \sqrt{70} \text{Eeu2}-2 \sqrt{42} \text{Meu}\right) & k=4\land (m=-4\lor m=4) \\ \frac{3}{28} \left(12 \text{Ea2u}-14 \text{Eb1u}-14 \text{Eb2u}+9 \text{Eeu1}+7 \text{Eeu2}-2 \sqrt{15} \text{Meu}\right) & k=4\land m=0 \\ -\frac{13}{560} \sqrt{3} \left(4 \sqrt{42} \text{Eb1u}-4 \sqrt{42} \text{Eb2u}+5 \sqrt{42} \text{Eeu1}-5 \sqrt{42} \text{Eeu2}+2 \sqrt{70} \text{Meu}\right) & k=6\land (m=-4\lor m=4) \\ \frac{13}{280} \left(40 \text{Ea2u}+12 \text{Eb1u}+12 \text{Eb2u}-25 \text{Eeu1}-39 \text{Eeu2}-14 \sqrt{15} \text{Meu}\right) & k=6\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea2u + Eb1u + Eb2u + 2*Eeu1 + 2*Eeu2)/7, k == 0 && m == 0}, {(5*(Ea2u - Eeu1 + Sqrt[15]*Meu))/7, k == 2 && m == 0}, {(-3*(2*Sqrt[70]*Eb1u - 2*Sqrt[70]*Eb2u - 3*Sqrt[70]*Eeu1 + 3*Sqrt[70]*Eeu2 - 2*Sqrt[42]*Meu))/56, k == 4 && (m == -4 || m == 4)}, {(3*(12*Ea2u - 14*Eb1u - 14*Eb2u + 9*Eeu1 + 7*Eeu2 - 2*Sqrt[15]*Meu))/28, k == 4 && m == 0}, {(-13*Sqrt[3]*(4*Sqrt[42]*Eb1u - 4*Sqrt[42]*Eb2u + 5*Sqrt[42]*Eeu1 - 5*Sqrt[42]*Eeu2 + 2*Sqrt[70]*Meu))/560, k == 6 && (m == -4 || m == 4)}, {(13*(40*Ea2u + 12*Eb1u + 12*Eb2u - 25*Eeu1 - 39*Eeu2 - 14*Sqrt[15]*Meu))/280, k == 6 && m == 0}}, 0] ### ### Akm = {{0, 0, (1/7)*(Ea2u + Eb1u + Eb2u + (2)*(Eeu1) + (2)*(Eeu2))} , {2, 0, (5/7)*(Ea2u + (-1)*(Eeu1) + (sqrt(15))*(Meu))} , {4, 0, (3/28)*((12)*(Ea2u) + (-14)*(Eb1u) + (-14)*(Eb2u) + (9)*(Eeu1) + (7)*(Eeu2) + (-2)*((sqrt(15))*(Meu)))} , {4,-4, (-3/56)*((2)*((sqrt(70))*(Eb1u)) + (-2)*((sqrt(70))*(Eb2u)) + (-3)*((sqrt(70))*(Eeu1)) + (3)*((sqrt(70))*(Eeu2)) + (-2)*((sqrt(42))*(Meu)))} , {4, 4, (-3/56)*((2)*((sqrt(70))*(Eb1u)) + (-2)*((sqrt(70))*(Eb2u)) + (-3)*((sqrt(70))*(Eeu1)) + (3)*((sqrt(70))*(Eeu2)) + (-2)*((sqrt(42))*(Meu)))} , {6, 0, (13/280)*((40)*(Ea2u) + (12)*(Eb1u) + (12)*(Eb2u) + (-25)*(Eeu1) + (-39)*(Eeu2) + (-14)*((sqrt(15))*(Meu)))} , {6,-4, (-13/560)*((sqrt(3))*((4)*((sqrt(42))*(Eb1u)) + (-4)*((sqrt(42))*(Eb2u)) + (5)*((sqrt(42))*(Eeu1)) + (-5)*((sqrt(42))*(Eeu2)) + (2)*((sqrt(70))*(Meu))))} , {6, 4, (-13/560)*((sqrt(3))*((4)*((sqrt(42))*(Eb1u)) + (-4)*((sqrt(42))*(Eb2u)) + (5)*((sqrt(42))*(Eeu1)) + (-5)*((sqrt(42))*(Eeu2)) + (2)*((sqrt(70))*(Meu))))} } ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{-3}^{(3)}} $|$ \frac{1}{8} \left(5 \text{Eeu1}+3 \text{Eeu2}-2 \sqrt{15} \text{Meu}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right) $|$ 0 $|$ 0 $| ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Eb1u}+\text{Eb2u}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eb2u}-\text{Eb1u}}{2} $|$ 0 $| ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(3 \text{Eeu1}+5 \text{Eeu2}+2 \sqrt{15} \text{Meu}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right) $| ^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea2u} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(3 \text{Eeu1}+5 \text{Eeu2}+2 \sqrt{15} \text{Meu}\right) $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{\text{Eb2u}-\text{Eb1u}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eb1u}+\text{Eb2u}}{2} $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(\sqrt{15} \text{Eeu1}-\sqrt{15} \text{Eeu2}+2 \text{Meu}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(5 \text{Eeu1}+3 \text{Eeu2}-2 \sqrt{15} \text{Meu}\right) $| ### ### | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ ^$ f_{\text{xyz}} $|$ \text{Eb1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $| ^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ -\text{Meu} $|$ 0 $| ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea2u} $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|$ 0 $| ^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ 0 $|$ -\text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb2u} $| ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| ^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| ^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| ^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| ### ### ^ ^$$\text{Eb1u}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | ^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: | ^ ^$$\text{Eeu1}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: | ^ ^$$\text{Ea2u}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | ^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: | ^ ^$$\text{Eeu2}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_6.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: | ^ ^$$\text{Eb2u}$$ | {{:physics_chemistry:pointgroup:d4h_zxy_orb_3_7.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | ### ===== Coupling between two shells ===== ### Click on one of the subsections to expand it or ### ==== Potential for s-d orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 2\lor m\neq 0 \\ A(2,0) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, A[2, 0]] ### ### Akm = {{2, 0, A(2,0)} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ 0 $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $| ### ### | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ ^$ \text{s} $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $| ### ==== Potential for p-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k=0\land m=0 \\ A(2,0) & k=2\land m=0 \\ A(4,4) & k=4\land (m=-4\lor m=4) \\ A(4,0) & k=4\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k == 0 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[4, 4], k == 4 && (m == -4 || m == 4)}, {A[4, 0], k == 4 && m == 0}}, 0] ### ### Akm = {{2, 0, A(2,0)} , {4, 0, A(4,0)} , {4,-4, A(4,4)} , {4, 4, A(4,4)} } ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ -\frac{2 A(4,4)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $| ### ### | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ ^$ p_x $|$ 0 $|$ \frac{5 A(4,0)-9 A(2,0)}{10 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} A(4,4) $|$ 0 $|$ 0 $|$ \frac{5 A(4,0)-9 A(2,0)}{6 \sqrt{35}}-\frac{A(4,4)}{3 \sqrt{2}} $|$ 0 $|$ 0 $| ^$ p_y $|$ 0 $|$ 0 $|$ \frac{5 A(4,0)-9 A(2,0)}{10 \sqrt{21}}+\frac{1}{3} \sqrt{\frac{5}{6}} A(4,4) $|$ 0 $|$ 0 $|$ \frac{9 A(2,0)-5 A(4,0)}{6 \sqrt{35}}+\frac{A(4,4)}{3 \sqrt{2}} $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 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]] | | | | | | ###