~~CLOSETOC~~ ====== Orientation Z ====== ===== Symmetry Operations ===== ### In the S4 Point Group, with orientation Z there are the following symmetry operations ### ### {{:physics_chemistry:pointgroup:s4_z.png}} ### ### ^ Operator ^ Orientation ^ ^ $\text{E}$ | $\{0,0,0\}$ , | ^ $S_4$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | ^ $C_2$ | $\{0,0,1\}$ , | ### ===== Different Settings ===== ### * [[physics_chemistry:point_groups:s4:orientation_z|Point Group S4 with orientation Z]] ### ===== Character Table ===== ### | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ S_4 \,{\text{(2)}} $ ^ $ C_2 \,{\text{(1)}} $ ^ ^ $ \text{A} $ | $ 1 $ | $ 1 $ | $ 1 $ | ^ $ \text{B} $ | $ 1 $ | $ -1 $ | $ 1 $ | ^ $ \text{E} $ | $ 2 $ | $ 0 $ | $ -2 $ | ### ===== Product Table ===== ### | $ $ ^ $ \text{A} $ ^ $ \text{B} $ ^ $ \text{E} $ ^ ^ $ \text{A} $ | $ \text{A} $ | $ \text{B} $ | $ \text{E} $ | ^ $ \text{B} $ | $ \text{B} $ | $ \text{A} $ | $ \text{E} $ | ^ $ \text{E} $ | $ \text{E} $ | $ \text{E} $ | $ 2 \text{A}+2 \text{B} $ | ### ===== Sub Groups with compatible settings ===== ### * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] * [[physics_chemistry:point_groups:c2:orientation_z|Point Group C2 with orientation Z]] ### ===== Super Groups with compatible settings ===== ### * [[physics_chemistry:point_groups:c4h:orientation_z|Point Group C4h with orientation Z]] * [[physics_chemistry:point_groups:d2d:orientation_zxy|Point Group D2d with orientation Zxy]] * [[physics_chemistry:point_groups:d4h:orientation_zxy|Point Group D4h with orientation Zxy]] * [[physics_chemistry:point_groups:oh:orientation_xyz|Point Group Oh with orientation XYZ]] * [[physics_chemistry:point_groups:td:orientation_xyz|Point Group Td 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 S4 Point group with orientation Z the form of the expansion coefficients is: ### ==== Expansion ==== ### $$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(2,0) & k=2\land m=0 \\ A(3,2)-i B(3,2) & k=3\land m=-2 \\ A(3,2)+i B(3,2) & k=3\land m=2 \\ A(4,4)-i B(4,4) & k=4\land m=-4 \\ A(4,0) & k=4\land m=0 \\ A(4,4)+i B(4,4) & k=4\land m=4 \\ A(5,2)-i B(5,2) & k=5\land m=-2 \\ A(5,2)+i B(5,2) & k=5\land m=2 \\ A(6,4)-i B(6,4) & k=6\land m=-4 \\ A(6,0) & k=6\land m=0 \\ A(6,4)+i B(6,4) & k=6\land m=4 \end{cases}$$ ### ==== 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[3, 2] - I*B[3, 2], k == 3 && m == -2}, {A[3, 2] + I*B[3, 2], k == 3 && m == 2}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 0], k == 4 && m == 0}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {A[5, 2] - I*B[5, 2], k == 5 && m == -2}, {A[5, 2] + I*B[5, 2], k == 5 && m == 2}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 0], k == 6 && m == 0}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}}, 0] ### ==== Input format suitable for Quanty ==== ### Akm = {{0, 0, A(0,0)} , {2, 0, A(2,0)} , {3,-2, A(3,2) + (-I)*(B(3,2))} , {3, 2, A(3,2) + (I)*(B(3,2))} , {4, 0, A(4,0)} , {4,-4, A(4,4) + (-I)*(B(4,4))} , {4, 4, A(4,4) + (I)*(B(4,4))} , {5,-2, A(5,2) + (-I)*(B(5,2))} , {5, 2, A(5,2) + (I)*(B(5,2))} , {6, 0, A(6,0)} , {6,-4, A(6,4) + (-I)*(B(6,4))} , {6, 4, A(6,4) + (I)*(B(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}{ \frac{\text{Asf}(3,2)+i \text{Bsf}(3,2)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,2)-i \text{Bsf}(3,2)}{\sqrt{7}} }$|$\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}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$\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)-i \text{Bpf}(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}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$ 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}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{2 (\text{Apf}(4,4)+i \text{Bpf}(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}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$\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)-i \text{Bdd}(4,4)) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)-i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)-i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)-i \text{Bpd}(3,2)) }$|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)+i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{\text{Adf}(3,2)-i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)-i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\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}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$| ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\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}{ -\frac{\text{Adf}(3,2)+i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)+i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)-i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$| ^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} \sqrt{3} (\text{Apd}(3,2)+i \text{Bpd}(3,2)) }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{10}{7}} (\text{Add}(4,4)+i \text{Bdd}(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}{ \frac{5}{33} (\text{Adf}(5,2)+i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)+i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{2 (\text{Apf}(4,4)-i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)-i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$|$\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)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(6,4)) $|$ 0 $|$ 0 $| ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,2)-i \text{Bsf}(3,2)}{\sqrt{7}} }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)-i \text{Bdf}(5,2)) }$|$\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)-i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)-i \text{Bff}(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}{ -\frac{\text{Adf}(3,2)-i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)-i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\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)-i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)-i \text{Bff}(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}{ \frac{5}{33} (\text{Adf}(5,2)+i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)+i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} (\text{Adf}(5,2)-i \text{Bdf}(5,2))-\frac{2 (\text{Adf}(3,2)-i \text{Bdf}(3,2))}{3 \sqrt{7}} }$|$ 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}{ -\frac{\text{Adf}(3,2)+i \text{Bdf}(3,2)}{\sqrt{21}}-\frac{5 (\text{Adf}(5,2)+i \text{Bdf}(5,2))}{11 \sqrt{3}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(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}{ \frac{\text{Asf}(3,2)+i \text{Bsf}(3,2)}{\sqrt{7}} }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{33} \sqrt{70} (\text{Aff}(4,4)+i \text{Bff}(4,4))+\frac{10}{143} \sqrt{14} (\text{Aff}(6,4)+i \text{Bff}(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)+i \text{Bpf}(4,4))}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,2)+i \text{Bdf}(3,2))-\frac{1}{33} \sqrt{5} (\text{Adf}(5,2)+i \text{Bdf}(5,2)) }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} (\text{Aff}(4,4)+i \text{Bff}(4,4))-\frac{5}{143} \sqrt{\frac{70}{3}} (\text{Aff}(6,4)+i \text{Bff}(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}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$| ^$ 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}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\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) $|$ \frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ 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}} $|$ \frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ 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}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\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 $|$ \frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \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}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$ 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}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$ \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 $|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$|$\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}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$| ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\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}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{\text{Adf}(3,2)}{3 \sqrt{14}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$|$\color{darkred}{ -\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$| ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ \frac{1}{7} \sqrt{6} \text{Apd}(3,2) }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$\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}{ \frac{\text{Adf}(3,2)}{3 \sqrt{14}}-\frac{5}{33} \sqrt{2} \text{Adf}(5,2) }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2) }$|$\color{darkred}{ -\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$| ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3,2) }$|$ -\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdd}(4,4) $|$ 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}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ f_{\text{xyz}} $|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3,2) }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{11} \sqrt{10} \text{Bdf}(5,2) }$|$\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 $|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $| ^$ 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) $|$ -\frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}} }$|$\color{darkred}{ \frac{\text{Adf}(3,2)}{3 \sqrt{14}}-\frac{5}{33} \sqrt{2} \text{Adf}(5,2) }$|$\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) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4) $|$ 0 $| ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{5}{6}} \text{Bpf}(4,4) $|$ -\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}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{\text{Adf}(3,2)}{3 \sqrt{14}} }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)+\frac{5 \text{Bdf}(5,2)}{33 \sqrt{2}} }$|$\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 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4) $|$ -\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}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,2)-\frac{5}{33} \sqrt{2} \text{Bdf}(5,2) }$|$ 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}} $|$ \frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$|$\color{darkred}{ -\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2) }$|$\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) $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4) $|$ 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 }$|$ \frac{\text{Bpf}(4,4)}{3 \sqrt{2}} $|$ \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}{ -\sqrt{\frac{5}{42}} \text{Adf}(3,2)-\frac{1}{11} \sqrt{\frac{10}{3}} \text{Adf}(5,2) }$|$\color{darkred}{ -\frac{1}{11} \sqrt{\frac{15}{2}} \text{Bdf}(5,2) }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{11} \sqrt{\frac{14}{3}} \text{Bff}(4,4)-\frac{5}{143} \sqrt{\frac{70}{3}} \text{Bff}(6,4) $|$ -\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}{ \sqrt{\frac{2}{7}} \text{Asf}(3,2) }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{11} \sqrt{10} \text{Adf}(5,2) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{33} \sqrt{70} \text{Bff}(4,4)-\frac{10}{143} \sqrt{14} \text{Bff}(6,4) $|$ 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 ==== ### $$A_{k,m} = \begin{cases} \text{Ea} & k=0\land m=0 \\ 0 & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{Ea, k == 0 && m == 0}}, 0] ### ### Akm = {{0, 0, Ea} } ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ \text{Ea} $| ### ### | $ $ ^ $ \text{s} $ ^ ^$ \text{s} $|$ \text{Ea} $| ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ ^$ \text{s} $|$ 1 $| ### ### ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:s4_z_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Eb}+2 \text{Ee}) & k=0\land m=0 \\ \frac{5 (\text{Eb}-\text{Ee})}{3} & k=2\land m=0 \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Eb + 2*Ee)/3, k == 0 && m == 0}, {(5*(Eb - Ee))/3, k == 2 && m == 0}}, 0] ### ### Akm = {{0, 0, (1/3)*(Eb + (2)*(Ee))} , {2, 0, (5/3)*(Eb + (-1)*(Ee))} } ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ \text{Ee} $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Eb} $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Ee} $| ### ### | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ ^$ p_x $|$ \text{Ee} $|$ 0 $|$ 0 $| ^$ p_y $|$ 0 $|$ \text{Ee} $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ \text{Eb} $| ### ### | $ $ ^ $ {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 $| ### ### ^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_1_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | ^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_1_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | ^ ^$$\text{Eb}$$ | {{:physics_chemistry:pointgroup:s4_z_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea}+\text{Eb1}+\text{Eb2}+2 \text{Ee}) & k=0\land m=0 \\ \text{Ea}-\text{Eb1}-\text{Eb2}+\text{Ee} & k=2\land m=0 \\ 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ -\frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eb1}-\text{Eb2}-2 i \text{Mb}) & k=4\land m=-4 \\ \frac{3}{10} (6 \text{Ea}+\text{Eb1}+\text{Eb2}-8 \text{Ee}) & k=4\land m=0 \\ -\frac{3}{2} \sqrt{\frac{7}{10}} (\text{Eb1}-\text{Eb2}+2 i \text{Mb}) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea + Eb1 + Eb2 + 2*Ee)/5, k == 0 && m == 0}, {Ea - Eb1 - Eb2 + Ee, k == 2 && m == 0}, {0, k != 4 || (m != -4 && m != 0 && m != 4)}, {(-3*Sqrt[7/10]*(Eb1 - Eb2 - (2*I)*Mb))/2, k == 4 && m == -4}, {(3*(6*Ea + Eb1 + Eb2 - 8*Ee))/10, k == 4 && m == 0}}, (-3*Sqrt[7/10]*(Eb1 - Eb2 + (2*I)*Mb))/2] ### ### Akm = {{0, 0, (1/5)*(Ea + Eb1 + Eb2 + (2)*(Ee))} , {2, 0, Ea + (-1)*(Eb1) + (-1)*(Eb2) + Ee} , {4, 0, (3/10)*((6)*(Ea) + Eb1 + Eb2 + (-8)*(Ee))} , {4,-4, (-3/2)*((sqrt(7/10))*(Eb1 + (-1)*(Eb2) + (-2*I)*(Mb)))} , {4, 4, (-3/2)*((sqrt(7/10))*(Eb1 + (-1)*(Eb2) + (2*I)*(Mb)))} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Eb1}+\text{Eb2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{2} (-\text{Eb1}+\text{Eb2}+2 i \text{Mb}) $| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Ee} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee} $|$ 0 $| ^$ {Y_{2}^{(2)}} $|$ \frac{1}{2} (-\text{Eb1}+\text{Eb2}-2 i \text{Mb}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Eb1}+\text{Eb2}}{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{Eb2} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb} $| ^$ d_{3z^2-r^2} $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Ee} $|$ 0 $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee} $|$ 0 $| ^$ d_{\text{xy}} $|$ \text{Mb} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb1} $| ### ### | $ $ ^ $ {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}} $| ### ### ^ ^$$\text{Eb2}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_2_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_2_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | ^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_2_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | ^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_2_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | ^ ^$$\text{Eb1}$$ | {{:physics_chemistry:pointgroup:s4_z_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea1}+\text{Ea2}+\text{Eb}+2 \text{Ee1}+2 \text{Ee2}) & k=0\land m=0 \\ \frac{5}{7} \left(\text{Eb}-\text{Ee1}+\sqrt{15} \text{Me1}\right) & k=2\land m=0 \\ 0 & (k\neq 4\land k\neq 6)\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ -\frac{3 \left(2 \sqrt{5} \text{Ea1}-2 \sqrt{5} \text{Ea2}-3 \sqrt{5} \text{Ee1}+3 \sqrt{5} \text{Ee2}-4 i \sqrt{5} \text{Ma}-2 \sqrt{3} \text{Me1}+8 i \sqrt{3} \text{Me2}\right)}{4 \sqrt{14}} & k=4\land m=-4 \\ -\frac{3}{28} \left(14 \text{Ea1}+14 \text{Ea2}-12 \text{Eb}-9 \text{Ee1}-7 \text{Ee2}+2 \sqrt{15} \text{Me1}\right) & k=4\land m=0 \\ -\frac{3 \left(2 \sqrt{5} \text{Ea1}-2 \sqrt{5} \text{Ea2}-3 \sqrt{5} \text{Ee1}+3 \sqrt{5} \text{Ee2}+4 i \sqrt{5} \text{Ma}-2 \sqrt{3} \text{Me1}-8 i \sqrt{3} \text{Me2}\right)}{4 \sqrt{14}} & k=4\land m=4 \\ -\frac{13 \left(12 \text{Ea1}-12 \text{Ea2}+15 \text{Ee1}-15 \text{Ee2}-24 i \text{Ma}+2 \sqrt{15} \text{Me1}-8 i \sqrt{15} \text{Me2}\right)}{40 \sqrt{14}} & k=6\land m=-4 \\ \frac{13}{280} \left(12 \text{Ea1}+12 \text{Ea2}+40 \text{Eb}-25 \text{Ee1}-39 \text{Ee2}-14 \sqrt{15} \text{Me1}\right) & k=6\land m=0 \\ -\frac{13 \left(12 \text{Ea1}-12 \text{Ea2}+15 \text{Ee1}-15 \text{Ee2}+24 i \text{Ma}+2 \sqrt{15} \text{Me1}+8 i \sqrt{15} \text{Me2}\right)}{40 \sqrt{14}} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea1 + Ea2 + Eb + 2*Ee1 + 2*Ee2)/7, k == 0 && m == 0}, {(5*(Eb - Ee1 + Sqrt[15]*Me1))/7, k == 2 && m == 0}, {0, (k != 4 && k != 6) || (m != -4 && m != 0 && m != 4)}, {(-3*(2*Sqrt[5]*Ea1 - 2*Sqrt[5]*Ea2 - 3*Sqrt[5]*Ee1 + 3*Sqrt[5]*Ee2 - (4*I)*Sqrt[5]*Ma - 2*Sqrt[3]*Me1 + (8*I)*Sqrt[3]*Me2))/(4*Sqrt[14]), k == 4 && m == -4}, {(-3*(14*Ea1 + 14*Ea2 - 12*Eb - 9*Ee1 - 7*Ee2 + 2*Sqrt[15]*Me1))/28, k == 4 && m == 0}, {(-3*(2*Sqrt[5]*Ea1 - 2*Sqrt[5]*Ea2 - 3*Sqrt[5]*Ee1 + 3*Sqrt[5]*Ee2 + (4*I)*Sqrt[5]*Ma - 2*Sqrt[3]*Me1 - (8*I)*Sqrt[3]*Me2))/(4*Sqrt[14]), k == 4 && m == 4}, {(-13*(12*Ea1 - 12*Ea2 + 15*Ee1 - 15*Ee2 - (24*I)*Ma + 2*Sqrt[15]*Me1 - (8*I)*Sqrt[15]*Me2))/(40*Sqrt[14]), k == 6 && m == -4}, {(13*(12*Ea1 + 12*Ea2 + 40*Eb - 25*Ee1 - 39*Ee2 - 14*Sqrt[15]*Me1))/280, k == 6 && m == 0}}, (-13*(12*Ea1 - 12*Ea2 + 15*Ee1 - 15*Ee2 + (24*I)*Ma + 2*Sqrt[15]*Me1 + (8*I)*Sqrt[15]*Me2))/(40*Sqrt[14])] ### ### Akm = {{0, 0, (1/7)*(Ea1 + Ea2 + Eb + (2)*(Ee1) + (2)*(Ee2))} , {2, 0, (5/7)*(Eb + (-1)*(Ee1) + (sqrt(15))*(Me1))} , {4, 0, (-3/28)*((14)*(Ea1) + (14)*(Ea2) + (-12)*(Eb) + (-9)*(Ee1) + (-7)*(Ee2) + (2)*((sqrt(15))*(Me1)))} , {4,-4, (-3/4)*((1/(sqrt(14)))*((2)*((sqrt(5))*(Ea1)) + (-2)*((sqrt(5))*(Ea2)) + (-3)*((sqrt(5))*(Ee1)) + (3)*((sqrt(5))*(Ee2)) + (-4*I)*((sqrt(5))*(Ma)) + (-2)*((sqrt(3))*(Me1)) + (8*I)*((sqrt(3))*(Me2))))} , {4, 4, (-3/4)*((1/(sqrt(14)))*((2)*((sqrt(5))*(Ea1)) + (-2)*((sqrt(5))*(Ea2)) + (-3)*((sqrt(5))*(Ee1)) + (3)*((sqrt(5))*(Ee2)) + (4*I)*((sqrt(5))*(Ma)) + (-2)*((sqrt(3))*(Me1)) + (-8*I)*((sqrt(3))*(Me2))))} , {6, 0, (13/280)*((12)*(Ea1) + (12)*(Ea2) + (40)*(Eb) + (-25)*(Ee1) + (-39)*(Ee2) + (-14)*((sqrt(15))*(Me1)))} , {6,-4, (-13/40)*((1/(sqrt(14)))*((12)*(Ea1) + (-12)*(Ea2) + (15)*(Ee1) + (-15)*(Ee2) + (-24*I)*(Ma) + (2)*((sqrt(15))*(Me1)) + (-8*I)*((sqrt(15))*(Me2))))} , {6, 4, (-13/40)*((1/(sqrt(14)))*((12)*(Ea1) + (-12)*(Ea2) + (15)*(Ee1) + (-15)*(Ee2) + (24*I)*(Ma) + (2)*((sqrt(15))*(Me1)) + (8*I)*((sqrt(15))*(Me2))))} } ### ### | $ $ ^ $ {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{Ee1}+3 \text{Ee2}-2 \sqrt{15} \text{Me1}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(\sqrt{15} \text{Ee1}-\sqrt{15} \text{Ee2}+2 \text{Me1}-8 i \text{Me2}\right) $|$ 0 $|$ 0 $| ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Ea1}+\text{Ea2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{2} (-\text{Ea1}+\text{Ea2}+2 i \text{Ma}) $|$ 0 $| ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(3 \text{Ee1}+5 \text{Ee2}+2 \sqrt{15} \text{Me1}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(\sqrt{15} \text{Ee1}-\sqrt{15} \text{Ee2}+2 \text{Me1}-8 i \text{Me2}\right) $| ^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \left(\sqrt{15} \text{Ee1}-\sqrt{15} \text{Ee2}+2 \text{Me1}+8 i \text{Me2}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(3 \text{Ee1}+5 \text{Ee2}+2 \sqrt{15} \text{Me1}\right) $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{1}{2} (-\text{Ea1}+\text{Ea2}-2 i \text{Ma}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ea1}+\text{Ea2}}{2} $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(\sqrt{15} \text{Ee1}-\sqrt{15} \text{Ee2}+2 \text{Me1}+8 i \text{Me2}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \left(5 \text{Ee1}+3 \text{Ee2}-2 \sqrt{15} \text{Me1}\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{Ea1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma} $| ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ \text{Me1} $|$ \text{Me2} $|$ 0 $| ^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ \text{Me2} $|$ -\text{Me1} $|$ 0 $| ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Eb} $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \text{Me1} $|$ \text{Me2} $|$ 0 $|$ \text{Ee2} $|$ 0 $|$ 0 $| ^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ \text{Me2} $|$ -\text{Me1} $|$ 0 $|$ 0 $|$ \text{Ee2} $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$ \text{Ma} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea2} $| ### ### | $ $ ^ $ {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 $| ### ### ^ ^$$\text{Ea1}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_3_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_3_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: | ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_3_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: | ^ ^$$\text{Eb}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_3_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | ^ ^$$\text{Ee2}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_3_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: | ^ ^$$\text{Ee2}$$ | {{:physics_chemistry:pointgroup:s4_z_orb_3_6.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: | ^ ^$$\text{Ea2}$$ | {{:physics_chemistry:pointgroup:s4_z_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 ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 2\lor m\neq 0 \\ \sqrt{5} \text{Ma} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 2 || m != 0}}, Sqrt[5]*Ma] ### ### Akm = {{2, 0, (sqrt(5))*(Ma)} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ 0 $|$ 0 $|$ \text{Ma} $|$ 0 $|$ 0 $| ### ### | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ ^$ \text{s} $|$ 0 $|$ \text{Ma} $|$ 0 $|$ 0 $|$ 0 $| ### ==== Potential for s-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ \sqrt{\frac{7}{2}} (\text{Ma2}+i \text{Ma1}) & k=3\land m=-2 \\ \sqrt{\frac{7}{2}} (\text{Ma2}-i \text{Ma1}) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 2)}, {Sqrt[7/2]*(I*Ma1 + Ma2), k == 3 && m == -2}}, Sqrt[7/2]*((-I)*Ma1 + Ma2)] ### ### Akm = {{3, 2, (sqrt(7/2))*((-I)*(Ma1) + Ma2)} , {3,-2, (sqrt(7/2))*((I)*(Ma1) + Ma2)} } ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{\text{Ma2}-i \text{Ma1}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ma2}+i \text{Ma1}}{\sqrt{2}} $|$ 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)} $ ^ ^$ \text{s} $|$ \text{Ma1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma2} $| ### ==== Potential for p-d orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ \frac{7 i (\text{Mb}+i \text{Me})}{\sqrt{6}} & k=3\land m=-2 \\ \frac{-7 \text{Me}-7 i \text{Mb}}{\sqrt{6}} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -2 && m != 2)}, {((7*I)*(Mb + I*Me))/Sqrt[6], k == 3 && m == -2}}, ((-7*I)*Mb - 7*Me)/Sqrt[6]] ### ### Akm = {{3, 2, (1/(sqrt(6)))*((-7*I)*(Mb) + (-7)*(Me))} , {3,-2, (7*I)*((1/(sqrt(6)))*(Mb + (I)*(Me)))} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Me}-i \text{Mb} $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ \frac{-\text{Me}-i \text{Mb}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i (\text{Mb}+i \text{Me})}{\sqrt{2}} $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ \text{Me}+i \text{Mb} $|$ 0 $|$ 0 $|$ 0 $| ### ### | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ ^$ p_x $|$ 0 $|$ 0 $|$ \text{Mb} $|$ -\text{Me} $|$ 0 $| ^$ p_y $|$ 0 $|$ 0 $|$ \text{Me} $|$ \text{Mb} $|$ 0 $| ^$ p_z $|$ -\text{Me} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb} $| ### ==== Potential for p-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ \frac{5}{21} \left(\sqrt{21} \text{Mb}-\sqrt{21} \text{Me1}-\sqrt{35} \text{Me2}\right) & k=2\land m=0 \\ \frac{3}{40} \left(5 \sqrt{30} \text{Me1}-15 \sqrt{2} \text{Me2}-8 i \sqrt{30} \text{Me3}\right) & k=4\land m=-4 \\ \frac{3}{28} \left(4 \sqrt{21} \text{Mb}+3 \sqrt{21} \text{Me1}+3 \sqrt{35} \text{Me2}\right) & k=4\land m=0 \\ \frac{3}{40} \left(5 \sqrt{30} \text{Me1}-15 \sqrt{2} \text{Me2}+8 i \sqrt{30} \text{Me3}\right) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || m != 0)) || (m != -4 && m != 0 && m != 4)}, {(5*(Sqrt[21]*Mb - Sqrt[21]*Me1 - Sqrt[35]*Me2))/21, k == 2 && m == 0}, {(3*(5*Sqrt[30]*Me1 - 15*Sqrt[2]*Me2 - (8*I)*Sqrt[30]*Me3))/40, k == 4 && m == -4}, {(3*(4*Sqrt[21]*Mb + 3*Sqrt[21]*Me1 + 3*Sqrt[35]*Me2))/28, k == 4 && m == 0}}, (3*(5*Sqrt[30]*Me1 - 15*Sqrt[2]*Me2 + (8*I)*Sqrt[30]*Me3))/40] ### ### Akm = {{2, 0, (5/21)*((sqrt(21))*(Mb) + (-1)*((sqrt(21))*(Me1)) + (-1)*((sqrt(35))*(Me2)))} , {4, 0, (3/28)*((4)*((sqrt(21))*(Mb)) + (3)*((sqrt(21))*(Me1)) + (3)*((sqrt(35))*(Me2)))} , {4,-4, (3/40)*((5)*((sqrt(30))*(Me1)) + (-15)*((sqrt(2))*(Me2)) + (-8*I)*((sqrt(30))*(Me3)))} , {4, 4, (3/40)*((5)*((sqrt(30))*(Me1)) + (-15)*((sqrt(2))*(Me2)) + (8*I)*((sqrt(30))*(Me3)))} } ### ### | $ $ ^ $ {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}{4} \left(-\sqrt{6} \text{Me1}-\sqrt{10} \text{Me2}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{20} \left(-5 \sqrt{10} \text{Me1}+5 \sqrt{6} \text{Me2}+8 i \sqrt{10} \text{Me3}\right) $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ \frac{1}{20} \left(-5 \sqrt{10} \text{Me1}+5 \sqrt{6} \text{Me2}-8 i \sqrt{10} \text{Me3}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{4} \left(-\sqrt{6} \text{Me1}-\sqrt{10} \text{Me2}\right) $|$ 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 $|$ \text{Me1} $|$ \text{Me3} $|$ 0 $|$ \text{Me2} $|$ \sqrt{\frac{3}{5}} \text{Me3} $|$ 0 $| ^$ p_y $|$ 0 $|$ -\text{Me3} $|$ \text{Me1} $|$ 0 $|$ \sqrt{\frac{3}{5}} \text{Me3} $|$ -\text{Me2} $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb} $|$ 0 $|$ 0 $|$ 0 $| ### ==== Potential for d-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 3\land k\neq 5)\lor (m\neq -2\land m\neq 2) \\ \frac{1}{2} \sqrt{\frac{7}{2}} \left(i \sqrt{5} \text{Ma1}+\sqrt{5} \text{Ma2}-3 (\text{Mb1}+i \text{Mb2})\right) & k=3\land m=-2 \\ \frac{1}{2} \sqrt{\frac{7}{2}} \left(-i \sqrt{5} \text{Ma1}+\sqrt{5} \text{Ma2}-3 \text{Mb1}+3 i \text{Mb2}\right) & k=3\land m=2 \\ \frac{11 (\text{Ma2}+i \text{Ma1})}{\sqrt{10}} & k=5\land m=-2 \\ \frac{11 (\text{Ma2}-i \text{Ma1})}{\sqrt{10}} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 3 && k != 5) || (m != -2 && m != 2)}, {(Sqrt[7/2]*(I*Sqrt[5]*Ma1 + Sqrt[5]*Ma2 - 3*(Mb1 + I*Mb2)))/2, k == 3 && m == -2}, {(Sqrt[7/2]*((-I)*Sqrt[5]*Ma1 + Sqrt[5]*Ma2 - 3*Mb1 + (3*I)*Mb2))/2, k == 3 && m == 2}, {(11*(I*Ma1 + Ma2))/Sqrt[10], k == 5 && m == -2}}, (11*((-I)*Ma1 + Ma2))/Sqrt[10]] ### ### Akm = {{3, 2, (1/2)*((sqrt(7/2))*((-I)*((sqrt(5))*(Ma1)) + (sqrt(5))*(Ma2) + (-3)*(Mb1) + (3*I)*(Mb2)))} , {3,-2, (1/2)*((sqrt(7/2))*((I)*((sqrt(5))*(Ma1)) + (sqrt(5))*(Ma2) + (-3)*(Mb1 + (I)*(Mb2))))} , {5, 2, (11)*((1/(sqrt(10)))*((-I)*(Ma1) + Ma2))} , {5,-2, (11)*((1/(sqrt(10)))*((I)*(Ma1) + Ma2))} } ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{-2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Mb1}+i \text{Mb2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{-1}^{(2)}} $|$ \frac{-i \text{Ma1}+\text{Ma2}-\sqrt{5} (\text{Mb1}-i \text{Mb2})}{2 \sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{2} \sqrt{\frac{3}{2}} \left(-i \sqrt{5} \text{Ma1}-\sqrt{5} \text{Ma2}+\text{Mb1}+i \text{Mb2}\right) $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ \frac{\text{Ma2}-i \text{Ma1}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ma2}+i \text{Ma1}}{\sqrt{2}} $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ \frac{1}{2} \sqrt{\frac{3}{2}} \left(i \sqrt{5} \text{Ma1}-\sqrt{5} \text{Ma2}+\text{Mb1}-i \text{Mb2}\right) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i \text{Ma1}+\text{Ma2}-\sqrt{5} (\text{Mb1}+i \text{Mb2})}{2 \sqrt{2}} $| ^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Mb1}-i \text{Mb2}}{\sqrt{2}} $|$ 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)} $ ^ ^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb1} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{3z^2-r^2} $|$ \text{Ma1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma2} $| ^$ d_{\text{yz}} $|$ 0 $|$ \text{Mb2}-\frac{\sqrt{5} \text{Ma1}}{2} $|$ \frac{1}{4} \left(\sqrt{5} \text{Ma2}+\text{Mb1}\right) $|$ 0 $|$ -\frac{\sqrt{3} \text{Ma1}}{2} $|$ \frac{1}{4} \sqrt{3} \left(\sqrt{5} \text{Mb1}-3 \text{Ma2}\right) $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{4} \left(-\sqrt{5} \text{Ma2}-\text{Mb1}\right) $|$ \text{Mb2}-\frac{\sqrt{5} \text{Ma1}}{2} $|$ 0 $|$ \frac{1}{4} \sqrt{3} \left(\sqrt{5} \text{Mb1}-3 \text{Ma2}\right) $|$ \frac{\sqrt{3} \text{Ma1}}{2} $|$ 0 $| ^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb2} $|$ 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]] | | | | | | ###