~~CLOSETOC~~ ====== Orientation Z ====== ===== Symmetry Operations ===== ### In the C5 Point Group, with orientation Z there are the following symmetry operations ### ### {{:physics_chemistry:pointgroup:c5_z.png}} ### ### ^ Operator ^ Orientation ^ ^ $\text{E}$ | $\{0,0,0\}$ , | ^ $C_5$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | ^ $C_5^2$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | ### ===== Different Settings ===== ### * [[physics_chemistry:point_groups:c5:orientation_x|Point Group C5 with orientation X]] * [[physics_chemistry:point_groups:c5:orientation_y|Point Group C5 with orientation Y]] * [[physics_chemistry:point_groups:c5:orientation_z|Point Group C5 with orientation Z]] ### ===== Character Table ===== ### | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_5 \,{\text{(2)}} $ ^ $ C_5^2{} \,{\text{(2)}} $ ^ ^ $ \text{A} $ | $ 1 $ | $ 1 $ | $ 1 $ | ^ $ E_1 $ | $ 2 $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | ^ $ E_2 $ | $ 2 $ | $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ | $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ | ### ===== Product Table ===== ### | $ $ ^ $ \text{A} $ ^ $ E_1 $ ^ $ E_2 $ ^ ^ $ \text{A} $ | $ \text{A} $ | $ E_1 $ | $ E_2 $ | ^ $ E_1 $ | $ E_1 $ | $ 2 \text{A}+E_2 $ | $ E_1+E_2 $ | ^ $ E_2 $ | $ E_2 $ | $ E_1+E_2 $ | $ 2 \text{A}+E_1 $ | ### ===== Sub Groups with compatible settings ===== ### * [[physics_chemistry:point_groups:c1:orientation_1|Point Group C1 with orientation 1]] ### ===== Super Groups with compatible settings ===== ### * [[physics_chemistry:point_groups:d5d:orientation_zx|Point Group D5d with orientation Zx]] * [[physics_chemistry:point_groups:d5d:orientation_zy|Point Group D5d with orientation Zy]] * [[physics_chemistry:point_groups:d5h:orientation_zx|Point Group D5h with orientation Zx]] * [[physics_chemistry:point_groups:d5h:orientation_zy|Point Group D5h with orientation Zy]] ### ===== 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 C5 Point group with orientation Z the form of the expansion coefficients is: ### ==== Expansion ==== ### $$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(1,0) & k=1\land m=0 \\ A(2,0) & k=2\land m=0 \\ A(3,0) & k=3\land m=0 \\ A(4,0) & k=4\land m=0 \\ -A(5,5)+i B(5,5) & k=5\land m=-5 \\ A(5,0) & k=5\land m=0 \\ A(5,5)+i B(5,5) & k=5\land m=5 \\ -A(6,5)+i B(6,5) & k=6\land m=-5 \\ A(6,0) & k=6\land m=0 \\ A(6,5)+i B(6,5) & k=6\land m=5 \end{cases}$$ ### ==== Input format suitable for Mathematica (Quanty.nb) ==== ### Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 0], k == 2 && m == 0}, {A[3, 0], k == 3 && m == 0}, {A[4, 0], k == 4 && m == 0}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {A[5, 0], k == 5 && m == 0}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {-A[6, 5] + I*B[6, 5], k == 6 && m == -5}, {A[6, 0], k == 6 && m == 0}, {A[6, 5] + I*B[6, 5], k == 6 && m == 5}}, 0] ### ==== Input format suitable for Quanty ==== ### Akm = {{0, 0, A(0,0)} , {1, 0, A(1,0)} , {2, 0, A(2,0)} , {3, 0, A(3,0)} , {4, 0, A(4,0)} , {5, 0, A(5,0)} , {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} , {5, 5, A(5,5) + (I)*(B(5,5))} , {6, 0, A(6,0)} , {6,-5, (-1)*(A(6,5)) + (I)*(B(6,5))} , {6, 5, A(6,5) + (I)*(B(6,5))} } ### ==== 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}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\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}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\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}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\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 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\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}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\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}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\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}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$ 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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {Y_{2}^{(2)}} $|$ 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) $|$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$| ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (-\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$|$ \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 $|$ 0 $|$ \frac{5}{13} \sqrt{\frac{14}{33}} (-\text{Aff}(6,5)+i \text{Bff}(6,5)) $|$ 0 $| ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\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 $|$ 0 $|$ -\frac{5}{13} \sqrt{\frac{14}{33}} (-\text{Aff}(6,5)+i \text{Bff}(6,5)) $| ^$ {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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\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 $|$ 0 $| ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ 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}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$ -\frac{5}{13} \sqrt{\frac{14}{33}} (\text{Aff}(6,5)+i \text{Bff}(6,5)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} (\text{Adf}(5,5)+i \text{Bdf}(5,5)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{5}{13} \sqrt{\frac{14}{33}} (\text{Aff}(6,5)+i \text{Bff}(6,5)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| ### ==== 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_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 $| ^$ 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 $| ^$ 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 }$| ^$ 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_{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{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_{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 }$| ^$ f_{y\left(3x^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 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $| ^$ 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_{y\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 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $| ^$ f_{z\left(5z^2-3r^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(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 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$ 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 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| ^$ f_{x\left(x^2-3y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| ### ==== One particle coupling on a basis of symmetry adapted functions ==== ### After rotation we find ### ### | $ $ ^ $ \text{s} $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ ^$ \text{s} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\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}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ p_y $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\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 $|$ 0 $| ^$ p_z $|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$ 0 $|$ \text{App}(0,0)+\frac{2}{5} \text{App}(2,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\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 $| ^$ p_x $|$\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}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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 $| ^$ d_{\text{xy}} $|$ 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 $|$ 0 $|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$| ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$\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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\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}{ \frac{2 \text{Apd}(1,0)}{\sqrt{15}}+\frac{3}{7} \sqrt{\frac{3}{5}} \text{Apd}(3,0) }$|$\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}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1,0)}{\sqrt{5}}-\frac{3 \text{Apd}(3,0)}{7 \sqrt{5}} }$|$ 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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ d_{x^2-y^2} $|$ 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) $|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$| ^$ f_{y\left(3x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|$ 0 $| ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Bff}(6,5) $| ^$ f_{y\left(5z^2-r^2\right)} $|$\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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\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 $|$ 0 $| ^$ f_{z\left(5z^2-3r^2\right)} $|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$ 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}{ \frac{3 \text{Adf}(1,0)}{\sqrt{35}}+\frac{4 \text{Adf}(3,0)}{3 \sqrt{35}}+\frac{10}{33} \sqrt{\frac{5}{7}} \text{Adf}(5,0) }$|$\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(5z^2-r^2\right)} $|$\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}{ 2 \sqrt{\frac{2}{35}} \text{Adf}(1,0)+\frac{1}{3} \sqrt{\frac{2}{35}} \text{Adf}(3,0)-\frac{5}{33} \sqrt{\frac{10}{7}} \text{Adf}(5,0) }$|$\color{darkred}{ 0 }$|$ 0 $|$ 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 $| ^$ 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}{ \frac{\text{Adf}(1,0)}{\sqrt{7}}-\frac{2 \text{Adf}(3,0)}{3 \sqrt{7}}+\frac{5 \text{Adf}(5,0)}{33 \sqrt{7}} }$|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ -\frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $| ^$ f_{x\left(x^2-3y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{5}{11} \sqrt{\frac{2}{3}} \text{Bdf}(5,5) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{5}{11} \sqrt{\frac{2}{3}} \text{Adf}(5,5) }$|$ 0 $|$ \frac{5}{13} \sqrt{\frac{14}{33}} \text{Bff}(6,5) $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{5}{13} \sqrt{\frac{14}{33}} \text{Aff}(6,5) $|$ \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) $| ### ===== 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:c5_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea}+2 \text{Ee1}) & k=0\land m=0 \\ 0 & k\neq 2\lor m\neq 0 \\ \frac{5 (\text{Ea}-\text{Ee1})}{3} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea + 2*Ee1)/3, k == 0 && m == 0}, {0, k != 2 || m != 0}}, (5*(Ea - Ee1))/3] ### ### Akm = {{0, 0, (1/3)*(Ea + (2)*(Ee1))} , {2, 0, (5/3)*(Ea + (-1)*(Ee1))} } ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ \text{Ee1} $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea} $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Ee1} $| ### ### | $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ ^$ p_y $|$ \text{Ee1} $|$ 0 $|$ 0 $| ^$ p_z $|$ 0 $|$ \text{Ea} $|$ 0 $| ^$ p_x $|$ 0 $|$ 0 $|$ \text{Ee1} $| ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| ### ### ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_1_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_1_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_1_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | ### ==== Potential for d orbitals ==== ### $$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea}+2 (\text{Ee1}+\text{Ee2})) & k=0\land m=0 \\ 0 & (k\neq 2\land k\neq 4)\lor m\neq 0 \\ \text{Ea}+\text{Ee1}-2 \text{Ee2} & k=2\land m=0 \\ \frac{3}{5} (3 \text{Ea}-4 \text{Ee1}+\text{Ee2}) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea + 2*(Ee1 + Ee2))/5, k == 0 && m == 0}, {0, (k != 2 && k != 4) || m != 0}, {Ea + Ee1 - 2*Ee2, k == 2 && m == 0}}, (3*(3*Ea - 4*Ee1 + Ee2))/5] ### ### Akm = {{0, 0, (1/5)*(Ea + (2)*(Ee1 + Ee2))} , {2, 0, Ea + Ee1 + (-2)*(Ee2)} , {4, 0, (3/5)*((3)*(Ea) + (-4)*(Ee1) + Ee2)} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-2}^{(2)}} $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| ^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2} $| ### ### | $ $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ ^$ d_{\text{xy}} $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{\text{yz}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| ^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2} $| ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| ### ### ^ ^$$\text{Ee2}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_2_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: | ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_2_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_2_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_2_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | ^ ^$$\text{Ee2}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_2_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | ### ==== Potential for f orbitals ==== ### $$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Ea}+2 (\text{Ee1}+\text{Ee2a}+\text{Ee2b})) & k=0\land m=0 \\ 0 & (k\neq 6\land ((k\neq 2\land k\neq 4)\lor m\neq 0))\lor (m\neq -5\land m\neq 0\land m\neq 5) \\ \frac{5}{14} (2 \text{Ea}+3 \text{Ee1}-5 \text{Ee2b}) & k=2\land m=0 \\ \frac{3}{7} (3 \text{Ea}+\text{Ee1}-7 \text{Ee2a}+3 \text{Ee2b}) & k=4\land m=0 \\ \frac{13}{5} i \sqrt{\frac{33}{14}} (\text{Me2Im}+i \text{Me2Re}) & k=6\land m=-5 \\ \frac{13}{70} (10 \text{Ea}-15 \text{Ee1}+6 \text{Ee2a}-\text{Ee2b}) & k=6\land m=0 \\ \frac{13}{5} \sqrt{\frac{33}{14}} (\text{Me2Re}+i \text{Me2Im}) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea + 2*(Ee1 + Ee2a + Ee2b))/7, k == 0 && m == 0}, {0, (k != 6 && ((k != 2 && k != 4) || m != 0)) || (m != -5 && m != 0 && m != 5)}, {(5*(2*Ea + 3*Ee1 - 5*Ee2b))/14, k == 2 && m == 0}, {(3*(3*Ea + Ee1 - 7*Ee2a + 3*Ee2b))/7, k == 4 && m == 0}, {((13*I)/5)*Sqrt[33/14]*(Me2Im + I*Me2Re), k == 6 && m == -5}, {(13*(10*Ea - 15*Ee1 + 6*Ee2a - Ee2b))/70, k == 6 && m == 0}}, (13*Sqrt[33/14]*(I*Me2Im + Me2Re))/5] ### ### Akm = {{0, 0, (1/7)*(Ea + (2)*(Ee1 + Ee2a + Ee2b))} , {2, 0, (5/14)*((2)*(Ea) + (3)*(Ee1) + (-5)*(Ee2b))} , {4, 0, (3/7)*((3)*(Ea) + Ee1 + (-7)*(Ee2a) + (3)*(Ee2b))} , {6, 0, (13/70)*((10)*(Ea) + (-15)*(Ee1) + (6)*(Ee2a) + (-1)*(Ee2b))} , {6,-5, (13/5*I)*((sqrt(33/14))*(Me2Im + (I)*(Me2Re)))} , {6, 5, (13/5)*((sqrt(33/14))*((I)*(Me2Im) + Me2Re))} } ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{-3}^{(3)}} $|$ \text{Ee2b} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\text{Me2Re}+i \text{Me2Im} $|$ 0 $| ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \text{Ee2a} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Me2Re}-i \text{Me2Im} $| ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$ -\text{Me2Re}-i \text{Me2Im} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2a} $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$ 0 $|$ \text{Me2Re}+i \text{Me2Im} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2b} $| ### ### | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ ^$ f_{y\left(3x^2-y^2\right)} $|$ \text{Ee2b} $|$ \text{Me2Re} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Me2Im} $|$ 0 $| ^$ f_{\text{xyz}} $|$ \text{Me2Re} $|$ \text{Ee2a} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Me2Im} $| ^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{z\left(5z^2-3r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$ \text{Me2Im} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2a} $|$ -\text{Me2Re} $| ^$ f_{x\left(x^2-3y^2\right)} $|$ 0 $|$ \text{Me2Im} $|$ 0 $|$ 0 $|$ 0 $|$ -\text{Me2Re} $|$ \text{Ee2b} $| ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ f_{y\left(3x^2-y^2\right)} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $| ^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| ^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $| ^$ f_{z\left(5z^2-3r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| ^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| ^$ f_{x\left(x^2-3y^2\right)} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| ### ### ^ ^$$\text{Ee2b}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_3_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$$ | ::: | ^ ^$$\text{Ee2a}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_3_2.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:c5_z_orb_3_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$$ | ::: | ^ ^$$\text{Ea}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_3_4.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | ^ ^$$\text{Ee1}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_3_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$$ | ::: | ^ ^$$\text{Ee2a}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_3_6.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | ^ ^$$\text{Ee2b}$$ | {{:physics_chemistry:pointgroup:c5_z_orb_3_7.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$$ | ::: | ### ===== Coupling between two shells ===== ### Click on one of the subsections to expand it or ### ==== Potential for s-p orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 1\lor m\neq 0 \\ A(1,0) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 1 || m != 0}}, A[1, 0]] ### ### Akm = {{1, 0, A(1,0)} } ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{A(1,0)}{\sqrt{3}} $|$ 0 $| ### ### | $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ ^$ \text{s} $|$ 0 $|$ \frac{A(1,0)}{\sqrt{3}} $|$ 0 $| ### ==== 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_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ ^$ \text{s} $|$ 0 $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $| ### ==== Potential for s-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 3\lor m\neq 0 \\ A(3,0) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 3 || m != 0}}, A[3, 0]] ### ### Akm = {{3, 0, A(3,0)} } ### ### | $ $ ^ $ {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 $|$ 0 $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ 0 $| ### ### | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ ^$ \text{s} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ 0 $| ### ==== Potential for p-d orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 1\land k\neq 3)\lor m\neq 0 \\ A(1,0) & k=1\land m=0 \\ A(3,0) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 1 && k != 3) || m != 0}, {A[1, 0], k == 1 && m == 0}}, A[3, 0]] ### ### Akm = {{1, 0, A(1,0)} , {3, 0, A(3,0)} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $| ### ### | $ $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ ^$ p_y $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $| ^$ p_x $|$ 0 $|$ 0 $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $| ### ==== Potential for p-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 2\land k\neq 4)\lor m\neq 0 \\ A(2,0) & k=2\land m=0 \\ A(4,0) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 2 && k != 4) || m != 0}, {A[2, 0], k == 2 && m == 0}}, A[4, 0]] ### ### Akm = {{2, 0, A(2,0)} , {4, 0, A(4,0)} } ### ### | $ $ ^ $ {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 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $| ### ### | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ ^$ p_y $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| ^$ p_x $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $| ### ==== Potential for d-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 5\land ((k\neq 1\land k\neq 3)\lor m\neq 0))\lor (m\neq -5\land m\neq 0\land m\neq 5) \\ A(1,0) & k=1\land m=0 \\ A(3,0) & k=3\land m=0 \\ -A(5,5)+i B(5,5) & k=5\land m=-5 \\ A(5,0) & k=5\land m=0 \\ A(5,5)+i B(5,5) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 5 && ((k != 1 && k != 3) || m != 0)) || (m != -5 && m != 0 && m != 5)}, {A[1, 0], k == 1 && m == 0}, {A[3, 0], k == 3 && m == 0}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {A[5, 0], k == 5 && m == 0}}, A[5, 5] + I*B[5, 5]] ### ### Akm = {{1, 0, A(1,0)} , {3, 0, A(3,0)} , {5, 0, A(5,0)} , {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} , {5, 5, A(5,5) + (I)*(B(5,5))} } ### ### | $ $ ^ $ {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 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i B(5,5)) $| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(2)}} $|$ -\frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)+i B(5,5)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $| ### ### | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ ^$ d_{\text{xy}} $|$ \frac{5}{11} \sqrt{\frac{2}{3}} A(5,5) $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{5}{11} \sqrt{\frac{2}{3}} B(5,5) $| ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ 0 $| ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $| ^$ d_{x^2-y^2} $|$ \frac{5}{11} \sqrt{\frac{2}{3}} B(5,5) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ -\frac{5}{11} \sqrt{\frac{2}{3}} A(5,5) $| ### ===== 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]] | | | | | | ###