~~CLOSETOC~~ ====== Orientation Z ====== ===== Symmetry Operations ===== ### In the C3 Point Group, with orientation Z there are the following symmetry operations ### ### {{:physics_chemistry:pointgroup:c3_z.png}} ### ### ^ Operator ^ Orientation ^ ^ $\text{E}$ | $\{0,0,0\}$ , | ^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , | ### ===== Different Settings ===== ### * [[physics_chemistry:point_groups:c3:orientation_x|Point Group C3 with orientation X]] * [[physics_chemistry:point_groups:c3:orientation_y|Point Group C3 with orientation Y]] * [[physics_chemistry:point_groups:c3:orientation_z|Point Group C3 with orientation Z]] ### ===== Character Table ===== ### | $ $ ^ $ \text{E} \,{\text{(1)}} $ ^ $ C_3 \,{\text{(2)}} $ ^ ^ $ \text{A} $ | $ 1 $ | $ 1 $ | ^ $ \text{E} $ | $ 2 $ | $ -1 $ | ### ===== Product Table ===== ### | $ $ ^ $ \text{A} $ ^ $ \text{E} $ ^ ^ $ \text{A} $ | $ \text{A} $ | $ \text{E} $ | ^ $ \text{E} $ | $ \text{E} $ | $ 2 \text{A}+\text{E} $ | ### ===== 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:c3h:orientation_z|Point Group C3h with orientation Z]] * [[physics_chemistry:point_groups:c3v:orientation_zx|Point Group C3v with orientation Zx]] * [[physics_chemistry:point_groups:c3v:orientation_zy|Point Group C3v with orientation Zy]] * [[physics_chemistry:point_groups:c6h:orientation_z|Point Group C6h with orientation Z]] * [[physics_chemistry:point_groups:c6v:orientation_zx|Point Group C6v with orientation Zx]] * [[physics_chemistry:point_groups:c6v:orientation_zy|Point Group C6v with orientation Zy]] * [[physics_chemistry:point_groups:c6:orientation_z|Point Group C6 with orientation Z]] * [[physics_chemistry:point_groups:d3d:orientation_zx|Point Group D3d with orientation Zx]] * [[physics_chemistry:point_groups:d3d:orientation_zx_a|Point Group D3d with orientation Zx_A]] * [[physics_chemistry:point_groups:d3d:orientation_zx_b|Point Group D3d with orientation Zx_B]] * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)|Point Group D3d with orientation Z(x-y)]] * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a|Point Group D3d with orientation Z(x-y)_A]] * [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b|Point Group D3d with orientation Z(x-y)_B]] * [[physics_chemistry:point_groups:d3d:orientation_zy|Point Group D3d with orientation Zy]] * [[physics_chemistry:point_groups:d3d:orientation_zy_a|Point Group D3d with orientation Zy_A]] * [[physics_chemistry:point_groups:d3d:orientation_zy_b|Point Group D3d with orientation Zy_B]] * [[physics_chemistry:point_groups:d3h:orientation_zx|Point Group D3h with orientation Zx]] * [[physics_chemistry:point_groups:d3h:orientation_zy|Point Group D3h with orientation Zy]] * [[physics_chemistry:point_groups:d3:orientation_zx|Point Group D3 with orientation Zx]] * [[physics_chemistry:point_groups:d3:orientation_zy|Point Group D3 with orientation Zy]] * [[physics_chemistry:point_groups:d6h:orientation_zx|Point Group D6h with orientation Zx]] * [[physics_chemistry:point_groups:d6h:orientation_zy|Point Group D6h with orientation Zy]] * [[physics_chemistry:point_groups:d6:orientation_zxy|Point Group D6 with orientation Zxy]] * [[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Point Group Oh with orientation 0sqrt2-1z]] * [[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Point Group Oh with orientation 0sqrt21z]] * [[physics_chemistry:point_groups:oh:orientation_111z|Point Group Oh with orientation 111z]] * [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Point Group Oh with orientation sqrt20-1z]] * [[physics_chemistry:point_groups:oh:orientation_sqrt201z|Point Group Oh with orientation sqrt201z]] * [[physics_chemistry:point_groups:s6:orientation_z|Point Group S6 with orientation Z]] ### ===== 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 C3 Point group with orientation Z the form of the expansion coefficients is: ### ==== Expansion ==== ### $$A_{k,m} = \begin{cases} A(0,0) & k=0\land m=0 \\ A(1,0) & k=1\land m=0 \\ A(2,0) & k=2\land m=0 \\ -A(3,3)+i B(3,3) & k=3\land m=-3 \\ A(3,0) & k=3\land m=0 \\ A(3,3)+i B(3,3) & k=3\land m=3 \\ -A(4,3)+i B(4,3) & k=4\land m=-3 \\ A(4,0) & k=4\land m=0 \\ A(4,3)+i B(4,3) & k=4\land m=3 \\ -A(5,3)+i B(5,3) & k=5\land m=-3 \\ A(5,0) & k=5\land m=0 \\ A(5,3)+i B(5,3) & k=5\land m=3 \\ A(6,6)-i B(6,6) & k=6\land m=-6 \\ -A(6,3)+i B(6,3) & k=6\land m=-3 \\ A(6,0) & k=6\land m=0 \\ A(6,3)+i B(6,3) & k=6\land m=3 \\ A(6,6)+i B(6,6) & k=6\land m=6 \end{cases}$$ ### ==== Input format suitable for Mathematica (Quanty.nb) ==== ### Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {A[1, 0], k == 1 && m == 0}, {A[2, 0], k == 2 && m == 0}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {A[3, 0], k == 3 && m == 0}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}, {A[4, 3] + I*B[4, 3], k == 4 && m == 3}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {A[5, 0], k == 5 && m == 0}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {-A[6, 3] + I*B[6, 3], k == 6 && m == -3}, {A[6, 0], k == 6 && m == 0}, {A[6, 3] + I*B[6, 3], k == 6 && m == 3}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 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)} , {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , {3, 3, A(3,3) + (I)*(B(3,3))} , {4, 0, A(4,0)} , {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , {4, 3, A(4,3) + (I)*(B(4,3))} , {5, 0, A(5,0)} , {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , {5, 3, A(5,3) + (I)*(B(5,3))} , {6, 0, A(6,0)} , {6,-3, (-1)*(A(6,3)) + (I)*(B(6,3))} , {6, 3, A(6,3) + (I)*(B(6,3))} , {6,-6, A(6,6) + (-I)*(B(6,6))} , {6, 6, A(6,6) + (I)*(B(6,6))} } ### ==== One particle coupling on a basis of spherical harmonics ==== ### The operator representing the potential in second quantisation is given as: $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice the integral for the expectation value in front of the creation and annihilation operators separates into a radial part and angular part. The angular part has an analytical solution, the radial integral is cast int a parameter. $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$ ### ### we can express the operator as $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ ### ### The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'',l'}(k,m)$ can be complex. Instead of allowing complex parameters we took $A_{l'',l'}(k,m) + \mathrm{I}\, B_{l'',l'}(k,m)$ (with both A and B real) as the expansion parameter. ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ \text{Ass}(0,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asp}(1,0)}{\sqrt{3}} }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{\text{Asd}(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$\color{darkred}{ -\frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$| ^$ {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}{ \frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{1}{3} (-\text{Apf}(4,3)+i \text{Bpf}(4,3)) $|$ 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 }$|$ -\frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ -\frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $| ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ \frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$|$\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 $|$ \frac{1}{3} (\text{Apf}(4,3)+i \text{Bpf}(4,3)) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $| ^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{3}{7} (-\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ 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}{ \frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ {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 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$\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}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\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}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\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}{ \frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$| ^$ {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}} }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))+\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\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}{ -\frac{3}{7} (\text{Apd}(3,3)+i \text{Bpd}(3,3)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Add}(4,3)+i \text{Bdd}(4,3)) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\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}{ \frac{-\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$|$ 0 $|$ \frac{-\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$|$\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 $|$ \frac{10}{143} \sqrt{\frac{7}{3}} (-\text{Aff}(6,3)+i \text{Bff}(6,3))-\frac{1}{11} \sqrt{7} (-\text{Aff}(4,3)+i \text{Bff}(4,3)) $|$ 0 $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)-i \text{Bff}(6,6)) $| ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{3} (\text{Apf}(4,3)-i \text{Bpf}(4,3)) $|$\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}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (-\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ -\frac{1}{33} \sqrt{14} (-\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{5}{143} \sqrt{42} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ 0 $|$ 0 $| ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 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}{ \frac{5}{33} \sqrt{2} (-\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (-\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{14} (-\text{Aff}(4,3)+i \text{Bff}(4,3))+\frac{5}{143} \sqrt{42} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ 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 }$|$ \frac{1}{11} \sqrt{7} (\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{10}{143} \sqrt{\frac{7}{3}} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{7} (-\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{10}{143} \sqrt{\frac{7}{3}} (-\text{Aff}(6,3)+i \text{Bff}(6,3)) $| ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$\color{darkred}{ \frac{5}{33} \sqrt{2} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$|$\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 $|$ \frac{1}{33} \sqrt{14} (\text{Aff}(4,3)+i \text{Bff}(4,3))+\frac{5}{143} \sqrt{42} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} (-\text{Apf}(4,3)-i \text{Bpf}(4,3)) $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3))-\frac{4}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3)) }$|$\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}} }$|$ 0 $|$ 0 $|$ -\frac{1}{33} \sqrt{14} (\text{Aff}(4,3)+i \text{Bff}(4,3))-\frac{5}{143} \sqrt{42} (\text{Aff}(6,3)+i \text{Bff}(6,3)) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3,3)+i \text{Bsf}(3,3)}{\sqrt{7}} }$|$ 0 $|$ \frac{\text{Apf}(4,3)+i \text{Bpf}(4,3)}{3 \sqrt{3}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{33} \sqrt{5} (\text{Adf}(5,3)+i \text{Bdf}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (\text{Adf}(3,3)+i \text{Bdf}(3,3)) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6,6)+i \text{Bff}(6,6)) $|$ 0 $|$ 0 $|$ \frac{10}{143} \sqrt{\frac{7}{3}} (\text{Aff}(6,3)+i \text{Bff}(6,3))-\frac{1}{11} \sqrt{7} (\text{Aff}(4,3)+i \text{Bff}(4,3)) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0) $| ### ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== ### Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field ### ### | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ p_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}{ \sqrt{\frac{2}{7}} \text{Bsf}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Asf}(3,0)}{\sqrt{7}} }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Asf}(3,3) }$| ^$ p_y $|$\color{darkred}{ 0 }$|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{3}{7} \text{Apd}(3,3) }$|$\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}{ \frac{3}{7} \text{Bpd}(3,3) }$|$ 0 $|$ \frac{1}{3} \text{Apf}(4,3) $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ 0 $|$ 0 $|$ \frac{1}{3} \text{Bpf}(4,3) $|$ 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 }$|$ \frac{1}{3} \sqrt{\frac{2}{3}} \text{Bpf}(4,3) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2,0)+\frac{4 \text{Apf}(4,0)}{3 \sqrt{21}} $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{3}} \text{Apf}(4,3) $| ^$ p_x $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0,0)-\frac{1}{5} \text{App}(2,0) $|$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$|$\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}{ -\frac{3}{7} \text{Apd}(3,3) }$|$ 0 $|$ \frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2,0)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4,0) $|$ -\frac{1}{3} \text{Apf}(4,3) $|$ 0 $| ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ \frac{3}{7} \text{Apd}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$|$ \text{Add}(0,0)-\frac{2}{7} \text{Add}(2,0)+\frac{1}{21} \text{Add}(4,0) $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 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}{33} \sqrt{2} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| ^$ 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 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\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}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\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}{ \frac{2}{33} \sqrt{10} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdf}(3,3) }$|$\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}{ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Adf}(3,3)-\frac{2}{33} \sqrt{10} \text{Adf}(5,3) }$| ^$ 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}} }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 0 $|$ 0 $|$ \text{Add}(0,0)+\frac{1}{7} \text{Add}(2,0)-\frac{4}{21} \text{Add}(4,0) $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\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}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$| ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ \frac{3}{7} \text{Bpd}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{3}{7} \text{Apd}(3,3) }$|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdd}(4,3) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4,3) $|$ \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}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$|$\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 }$| ^$ f_{y\left(3x^2-y^2\right)} $|$\color{darkred}{ \sqrt{\frac{2}{7}} \text{Bsf}(3,3) }$|$ 0 $|$ \frac{1}{3} \sqrt{\frac{2}{3}} \text{Bpf}(4,3) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{2}{33} \sqrt{10} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{10}{7}} \text{Bdf}(3,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)-\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $|$ 0 $|$ 0 $|$ \frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)-\frac{1}{11} \sqrt{14} \text{Bff}(4,3) $|$ 0 $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $| ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} \text{Apf}(4,3) $|$ 0 $|$ \frac{1}{3} \text{Bpf}(4,3) $|$\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{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)+\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,3)+\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|$ 0 $| ^$ 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}{ \frac{5}{33} \sqrt{2} \text{Adf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3) }$|$\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}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Aff}(4,3)+\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ \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 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 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 }$|$ \frac{10}{143} \sqrt{\frac{14}{3}} \text{Bff}(6,3)-\frac{1}{11} \sqrt{14} \text{Bff}(4,3) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{4}{15} \text{Aff}(2,0)+\frac{2}{11} \text{Aff}(4,0)+\frac{100}{429} \text{Aff}(6,0) $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{14} \text{Aff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,3) $| ^$ 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}{ \frac{5}{33} \sqrt{2} \text{Bdf}(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} \text{Bdf}(3,3) }$|$\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}{ \frac{1}{3} \sqrt{\frac{2}{7}} \text{Adf}(3,3)-\frac{5}{33} \sqrt{2} \text{Adf}(5,3) }$|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)+\frac{1}{5} \text{Aff}(2,0)+\frac{1}{33} \text{Aff}(4,0)-\frac{25}{143} \text{Aff}(6,0) $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,3)-\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{1}{3} \text{Bpf}(4,3) $|$ 0 $|$ -\frac{1}{3} \text{Apf}(4,3) $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Bdf}(3,3)+\frac{4}{33} \sqrt{5} \text{Bdf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Adf}(3,3)-\frac{4}{33} \sqrt{5} \text{Adf}(5,3) }$|$\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}} }$|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{14} \text{Bff}(4,3)+\frac{5}{143} \sqrt{42} \text{Bff}(6,3) $|$ 0 $|$ -\frac{1}{33} \sqrt{14} \text{Aff}(4,3)-\frac{5}{143} \sqrt{42} \text{Aff}(6,3) $|$ \text{Aff}(0,0)-\frac{7}{33} \text{Aff}(4,0)+\frac{10}{143} \text{Aff}(6,0) $|$ 0 $| ^$ f_{x\left(x^2-3y^2\right)} $|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Asf}(3,3) }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{3}} \text{Apf}(4,3) $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{3} \sqrt{\frac{10}{7}} \text{Adf}(3,3)-\frac{2}{33} \sqrt{10} \text{Adf}(5,3) }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6,6) $|$ 0 $|$ 0 $|$ \frac{1}{11} \sqrt{14} \text{Aff}(4,3)-\frac{10}{143} \sqrt{\frac{14}{3}} \text{Aff}(6,3) $|$ 0 $|$ 0 $|$ \text{Aff}(0,0)-\frac{1}{3} \text{Aff}(2,0)+\frac{1}{11} \text{Aff}(4,0)-\frac{5}{429} \text{Aff}(6,0)+\frac{10}{13} \sqrt{\frac{7}{33}} \text{Aff}(6,6) $| ### ===== 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:c3_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{3} (\text{Ea}+2 \text{Ee}) & k=0\land m=0 \\ 0 & k\neq 2\lor m\neq 0 \\ \frac{5 (\text{Ea}-\text{Ee})}{3} & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea + 2*Ee)/3, k == 0 && m == 0}, {0, k != 2 || m != 0}}, (5*(Ea - Ee))/3] ### ### Akm = {{0, 0, (1/3)*(Ea + (2)*(Ee))} , {2, 0, (5/3)*(Ea + (-1)*(Ee))} } ### ### | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ ^$ {Y_{-1}^{(1)}} $|$ \text{Ee} $|$ 0 $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea} $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Ee} $| ### ### | $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ ^$ p_y $|$ \text{Ee} $|$ 0 $|$ 0 $| ^$ p_z $|$ 0 $|$ \text{Ea} $|$ 0 $| ^$ p_x $|$ 0 $|$ 0 $|$ \text{Ee} $| ### ### | $ $ ^ $ {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}} $| ### ### ^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:c3_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:c3_z_orb_1_2.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | ^ ^$$\text{Ee}$$ | {{:physics_chemistry:pointgroup:c3_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{5} (\text{Ea}+2 (\text{Ee1}+\text{Ee2})) & k=0\land m=0 \\ 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ \text{Ea}+\text{Ee1}-2 \text{Ee2} & k=2\land m=0 \\ 3 i \sqrt{\frac{7}{5}} (\text{MeIm}+i \text{MeRe}) & k=4\land m=-3 \\ \frac{3}{5} (3 \text{Ea}-4 \text{Ee1}+\text{Ee2}) & k=4\land m=0 \\ 3 \sqrt{\frac{7}{5}} (\text{MeRe}+i \text{MeIm}) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Ea + 2*(Ee1 + Ee2))/5, k == 0 && m == 0}, {0, (k != 4 && (k != 2 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {Ea + Ee1 - 2*Ee2, k == 2 && m == 0}, {(3*I)*Sqrt[7/5]*(MeIm + I*MeRe), k == 4 && m == -3}, {(3*(3*Ea - 4*Ee1 + Ee2))/5, k == 4 && m == 0}}, 3*Sqrt[7/5]*(I*MeIm + MeRe)] ### ### Akm = {{0, 0, (1/5)*(Ea + (2)*(Ee1 + Ee2))} , {2, 0, Ea + Ee1 + (-2)*(Ee2)} , {4, 0, (3/5)*((3)*(Ea) + (-4)*(Ee1) + Ee2)} , {4,-3, (3*I)*((sqrt(7/5))*(MeIm + (I)*(MeRe)))} , {4, 3, (3)*((sqrt(7/5))*((I)*(MeIm) + MeRe))} } ### ### | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ ^$ {Y_{-2}^{(2)}} $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ \text{MeRe}-i \text{MeIm} $|$ 0 $| ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ -\text{MeRe}+i \text{MeIm} $| ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(2)}} $|$ \text{MeRe}+i \text{MeIm} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| ^$ {Y_{2}^{(2)}} $|$ 0 $|$ -\text{MeRe}-i \text{MeIm} $|$ 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} $|$ \text{MeRe} $|$ 0 $|$ \text{MeIm} $|$ 0 $| ^$ d_{\text{yz}} $|$ \text{MeRe} $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ \text{MeIm} $| ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| ^$ d_{\text{xz}} $|$ \text{MeIm} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ -\text{MeRe} $| ^$ d_{x^2-y^2} $|$ 0 $|$ \text{MeIm} $|$ 0 $|$ -\text{MeRe} $|$ \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:c3_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:c3_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:c3_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:c3_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:c3_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 ==== ### $$A_{k,m} = \begin{cases} \frac{1}{7} (\text{Eaxx23y2}+\text{Eaz3}+\text{Eaz3x2y2}+2 \text{Ee1}+2 \text{Ee2}) & k=0\land m=0 \\ 0 & (k\neq 6\land (((k\neq 2\lor m\neq 0)\land k\neq 4)\lor (m\neq -3\land m\neq 0\land m\neq 3)))\lor (m\neq -6\land m\neq -3\land m\neq 0\land m\neq 3\land m\neq 6) \\ -\frac{5}{28} (5 \text{Eaxx23y2}-4 \text{Eaz3}+5 \text{Eaz3x2y2}-6 \text{Ee1}) & k=2\land m=0 \\ -\frac{3 (3 \text{Maxx2y2z3}+3 i \text{Maz3y3x2y2}-2 i \text{MeIm}+2 \text{MeRe})}{\sqrt{14}} & k=4\land m=-3 \\ \frac{3}{14} (3 \text{Eaxx23y2}+6 \text{Eaz3}+3 \text{Eaz3x2y2}+2 \text{Ee1}-14 \text{Ee2}) & k=4\land m=0 \\ \frac{9 \text{Maxx2y2z3}-9 i \text{Maz3y3x2y2}+6 i \text{MeIm}+6 \text{MeRe}}{\sqrt{14}} & k=4\land m=3 \\ \frac{13}{20} \sqrt{\frac{33}{7}} (\text{Eaxx23y2}-\text{Eaz3x2y2}+2 i \text{Maxx23y2y3x2y2}) & k=6\land m=-6 \\ \frac{13}{5} \sqrt{\frac{3}{14}} (\text{Maxx2y2z3}+i (\text{Maz3y3x2y2}+3 \text{MeIm}+3 i \text{MeRe})) & k=6\land m=-3 \\ -\frac{13}{140} (\text{Eaxx23y2}-20 \text{Eaz3}+\text{Eaz3x2y2}+30 \text{Ee1}-12 \text{Ee2}) & k=6\land m=0 \\ \frac{13}{5} \sqrt{\frac{3}{14}} (-\text{Maxx2y2z3}+i \text{Maz3y3x2y2}+3 i \text{MeIm}+3 \text{MeRe}) & k=6\land m=3 \\ \frac{13}{20} \sqrt{\frac{33}{7}} (\text{Eaxx23y2}-\text{Eaz3x2y2}-2 i \text{Maxx23y2y3x2y2}) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{(Eaxx23y2 + Eaz3 + Eaz3x2y2 + 2*Ee1 + 2*Ee2)/7, k == 0 && m == 0}, {0, (k != 6 && (((k != 2 || m != 0) && k != 4) || (m != -3 && m != 0 && m != 3))) || (m != -6 && m != -3 && m != 0 && m != 3 && m != 6)}, {(-5*(5*Eaxx23y2 - 4*Eaz3 + 5*Eaz3x2y2 - 6*Ee1))/28, k == 2 && m == 0}, {(-3*(3*Maxx2y2z3 + (3*I)*Maz3y3x2y2 - (2*I)*MeIm + 2*MeRe))/Sqrt[14], k == 4 && m == -3}, {(3*(3*Eaxx23y2 + 6*Eaz3 + 3*Eaz3x2y2 + 2*Ee1 - 14*Ee2))/14, k == 4 && m == 0}, {(9*Maxx2y2z3 - (9*I)*Maz3y3x2y2 + (6*I)*MeIm + 6*MeRe)/Sqrt[14], k == 4 && m == 3}, {(13*Sqrt[33/7]*(Eaxx23y2 - Eaz3x2y2 + (2*I)*Maxx23y2y3x2y2))/20, k == 6 && m == -6}, {(13*Sqrt[3/14]*(Maxx2y2z3 + I*(Maz3y3x2y2 + 3*MeIm + (3*I)*MeRe)))/5, k == 6 && m == -3}, {(-13*(Eaxx23y2 - 20*Eaz3 + Eaz3x2y2 + 30*Ee1 - 12*Ee2))/140, k == 6 && m == 0}, {(13*Sqrt[3/14]*(-Maxx2y2z3 + I*Maz3y3x2y2 + (3*I)*MeIm + 3*MeRe))/5, k == 6 && m == 3}}, (13*Sqrt[33/7]*(Eaxx23y2 - Eaz3x2y2 - (2*I)*Maxx23y2y3x2y2))/20] ### ### Akm = {{0, 0, (1/7)*(Eaxx23y2 + Eaz3 + Eaz3x2y2 + (2)*(Ee1) + (2)*(Ee2))} , {2, 0, (-5/28)*((5)*(Eaxx23y2) + (-4)*(Eaz3) + (5)*(Eaz3x2y2) + (-6)*(Ee1))} , {4, 0, (3/14)*((3)*(Eaxx23y2) + (6)*(Eaz3) + (3)*(Eaz3x2y2) + (2)*(Ee1) + (-14)*(Ee2))} , {4, 3, (1/(sqrt(14)))*((9)*(Maxx2y2z3) + (-9*I)*(Maz3y3x2y2) + (6*I)*(MeIm) + (6)*(MeRe))} , {4,-3, (-3)*((1/(sqrt(14)))*((3)*(Maxx2y2z3) + (3*I)*(Maz3y3x2y2) + (-2*I)*(MeIm) + (2)*(MeRe)))} , {6, 0, (-13/140)*(Eaxx23y2 + (-20)*(Eaz3) + Eaz3x2y2 + (30)*(Ee1) + (-12)*(Ee2))} , {6, 3, (13/5)*((sqrt(3/14))*((-1)*(Maxx2y2z3) + (I)*(Maz3y3x2y2) + (3*I)*(MeIm) + (3)*(MeRe)))} , {6,-3, (13/5)*((sqrt(3/14))*(Maxx2y2z3 + (I)*(Maz3y3x2y2 + (3)*(MeIm) + (3*I)*(MeRe))))} , {6, 6, (13/20)*((sqrt(33/7))*(Eaxx23y2 + (-1)*(Eaz3x2y2) + (-2*I)*(Maxx23y2y3x2y2)))} , {6,-6, (13/20)*((sqrt(33/7))*(Eaxx23y2 + (-1)*(Eaz3x2y2) + (2*I)*(Maxx23y2y3x2y2)))} } ### ### | $ $ ^ $ {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{\text{Eaxx23y2}+\text{Eaz3x2y2}}{2} $|$ 0 $|$ 0 $|$ \frac{\text{Maxx2y2z3}+i \text{Maz3y3x2y2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{2} (-\text{Eaxx23y2}+\text{Eaz3x2y2}-2 i \text{Maxx23y2y3x2y2}) $| ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ \text{MeRe}-i \text{MeIm} $|$ 0 $|$ 0 $| ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ -\text{MeRe}+i \text{MeIm} $|$ 0 $| ^$ {Y_{0}^{(3)}} $|$ \frac{\text{Maxx2y2z3}-i \text{Maz3y3x2y2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \text{Eaz3} $|$ 0 $|$ 0 $|$ -\frac{\text{Maxx2y2z3}+i \text{Maz3y3x2y2}}{\sqrt{2}} $| ^$ {Y_{1}^{(3)}} $|$ 0 $|$ \text{MeRe}+i \text{MeIm} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $| ^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ -\text{MeRe}-i \text{MeIm} $|$ 0 $|$ 0 $|$ \text{Ee2} $|$ 0 $| ^$ {Y_{3}^{(3)}} $|$ \frac{1}{2} (-\text{Eaxx23y2}+\text{Eaz3x2y2}+2 i \text{Maxx23y2y3x2y2}) $|$ 0 $|$ 0 $|$ -\frac{\text{Maxx2y2z3}-i \text{Maz3y3x2y2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{\text{Eaxx23y2}+\text{Eaz3x2y2}}{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)} $ ^ ^$ f_{y\left(3x^2-y^2\right)} $|$ \text{Eaz3x2y2} $|$ 0 $|$ 0 $|$ \text{Maz3y3x2y2} $|$ 0 $|$ 0 $|$ \text{Maxx23y2y3x2y2} $| ^$ f_{\text{xyz}} $|$ 0 $|$ \text{Ee2} $|$ \text{MeRe} $|$ 0 $|$ \text{MeIm} $|$ 0 $|$ 0 $| ^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ \text{MeRe} $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ \text{MeIm} $|$ 0 $| ^$ f_{z\left(5z^2-3r^2\right)} $|$ \text{Maz3y3x2y2} $|$ 0 $|$ 0 $|$ \text{Eaz3} $|$ 0 $|$ 0 $|$ \text{Maxx2y2z3} $| ^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ \text{MeIm} $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ -\text{MeRe} $|$ 0 $| ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ \text{MeIm} $|$ 0 $|$ -\text{MeRe} $|$ \text{Ee2} $|$ 0 $| ^$ f_{x\left(x^2-3y^2\right)} $|$ \text{Maxx23y2y3x2y2} $|$ 0 $|$ 0 $|$ \text{Maxx2y2z3} $|$ 0 $|$ 0 $|$ \text{Eaxx23y2} $| ### ### | $ $ ^ $ {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}} $| ### ### ^ ^$$\text{Eaz3x2y2}$$ | {{:physics_chemistry:pointgroup:c3_z_orb_3_1.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$$ | ::: | ^ ^$$\text{Ee2}$$ | {{:physics_chemistry:pointgroup:c3_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:c3_z_orb_3_3.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$$ | ::: | ^ ^$$\text{Eaz3}$$ | {{:physics_chemistry:pointgroup:c3_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:c3_z_orb_3_5.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$$ | ::: | ^ ^$$\text{Ee2}$$ | {{:physics_chemistry:pointgroup:c3_z_orb_3_6.png?150}} | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | ^ ^$$\text{Eaxx23y2}$$ | {{:physics_chemistry:pointgroup:c3_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 ==== ### $$A_{k,m} = \begin{cases} 0 & k\neq 3\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ -A(3,3)+i B(3,3) & k=3\land m=-3 \\ A(3,0) & k=3\land m=0 \\ A(3,3)+i B(3,3) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, k != 3 || (m != -3 && m != 0 && m != 3)}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {A[3, 0], k == 3 && m == 0}}, A[3, 3] + I*B[3, 3]] ### ### Akm = {{3, 0, A(3,0)} , {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , {3, 3, A(3,3) + (I)*(B(3,3))} } ### ### | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ ^$ {Y_{0}^{(0)}} $|$ -\frac{A(3,3)+i B(3,3)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ \frac{A(3,3)-i B(3,3)}{\sqrt{7}} $| ### ### | $ $ ^ $ 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} $|$ \sqrt{\frac{2}{7}} B(3,3) $|$ 0 $|$ 0 $|$ \frac{A(3,0)}{\sqrt{7}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{7}} A(3,3) $| ### ==== Potential for p-d orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 3\land (k\neq 1\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ A(1,0) & k=1\land m=0 \\ -A(3,3)+i B(3,3) & k=3\land m=-3 \\ A(3,0) & k=3\land m=0 \\ A(3,3)+i B(3,3) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 3 && (k != 1 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {A[1, 0], k == 1 && m == 0}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {A[3, 0], k == 3 && m == 0}}, A[3, 3] + I*B[3, 3]] ### ### Akm = {{1, 0, A(1,0)} , {3, 0, A(3,0)} , {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , {3, 3, A(3,3) + (I)*(B(3,3))} } ### ### | $ $ ^ $ {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 $|$ -\frac{3}{7} (A(3,3)-i B(3,3)) $| ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $| ^$ {Y_{1}^{(1)}} $|$ \frac{3}{7} (A(3,3)+i B(3,3)) $|$ 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 $|$ \frac{3}{7} A(3,3) $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ 0 $|$ \frac{3}{7} B(3,3) $| ^$ p_z $|$ 0 $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $| ^$ p_x $|$ \frac{3}{7} B(3,3) $|$ 0 $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ -\frac{3}{7} A(3,3) $| ### ==== Potential for p-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ A(2,0) & k=2\land m=0 \\ -A(4,3)+i B(4,3) & k=4\land m=-3 \\ A(4,0) & k=4\land m=0 \\ A(4,3)+i B(4,3) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 4 && (k != 2 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {A[2, 0], k == 2 && m == 0}, {-A[4, 3] + I*B[4, 3], k == 4 && m == -3}, {A[4, 0], k == 4 && m == 0}}, A[4, 3] + I*B[4, 3]] ### ### Akm = {{2, 0, A(2,0)} , {4, 0, A(4,0)} , {4,-3, (-1)*(A(4,3)) + (I)*(B(4,3))} , {4, 3, A(4,3) + (I)*(B(4,3))} } ### ### | $ $ ^ $ {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 $|$ \frac{1}{3} (-A(4,3)+i B(4,3)) $|$ 0 $| ^$ {Y_{0}^{(1)}} $|$ -\frac{A(4,3)+i B(4,3)}{3 \sqrt{3}} $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ \frac{A(4,3)-i B(4,3)}{3 \sqrt{3}} $| ^$ {Y_{1}^{(1)}} $|$ 0 $|$ \frac{1}{3} (A(4,3)+i B(4,3)) $|$ 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 $|$ \frac{1}{3} A(4,3) $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $|$ \frac{1}{3} B(4,3) $|$ 0 $| ^$ p_z $|$ \frac{1}{3} \sqrt{\frac{2}{3}} B(4,3) $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{3}} A(4,3) $| ^$ p_x $|$ 0 $|$ \frac{1}{3} B(4,3) $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ -\frac{1}{3} A(4,3) $|$ 0 $| ### ==== Potential for d-f orbital mixing ==== ### $$A_{k,m} = \begin{cases} 0 & (k\neq 3\land k\neq 5\land (k\neq 1\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ A(1,0) & k=1\land m=0 \\ -A(3,3)+i B(3,3) & k=3\land m=-3 \\ A(3,0) & k=3\land m=0 \\ A(3,3)+i B(3,3) & k=3\land m=3 \\ -A(5,3)+i B(5,3) & k=5\land m=-3 \\ A(5,0) & k=5\land m=0 \\ A(5,3)+i B(5,3) & \text{True} \end{cases}$$ ### ### Akm[k_,m_]:=Piecewise[{{0, (k != 3 && k != 5 && (k != 1 || m != 0)) || (m != -3 && m != 0 && m != 3)}, {A[1, 0], k == 1 && m == 0}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {A[3, 0], k == 3 && m == 0}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {A[5, 0], k == 5 && m == 0}}, A[5, 3] + I*B[5, 3]] ### ### Akm = {{1, 0, A(1,0)} , {3, 0, A(3,0)} , {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , {3, 3, A(3,3) + (I)*(B(3,3))} , {5, 0, A(5,0)} , {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , {5, 3, A(5,3) + (I)*(B(5,3))} } ### ### | $ $ ^ $ {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 $|$ \frac{5}{33} \sqrt{2} (A(5,3)-i B(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)-i B(3,3)) $|$ 0 $|$ 0 $| ^$ {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 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i B(3,3))-\frac{4}{33} \sqrt{5} (A(5,3)-i B(5,3)) $|$ 0 $| ^$ {Y_{0}^{(2)}} $|$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i B(3,3))-\frac{2}{33} \sqrt{5} (A(5,3)+i B(5,3)) $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ \frac{2}{33} \sqrt{5} (A(5,3)-i B(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i B(3,3)) $| ^$ {Y_{1}^{(2)}} $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)+i B(3,3))+\frac{4}{33} \sqrt{5} (A(5,3)+i B(5,3)) $|$ 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)}} $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)+i B(3,3))-\frac{5}{33} \sqrt{2} (A(5,3)+i B(5,3)) $|$ 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}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ \frac{5}{33} \sqrt{2} A(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} A(3,3) $|$ 0 $|$ \frac{5}{33} \sqrt{2} B(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} B(3,3) $|$ 0 $|$ 0 $| ^$ d_{\text{yz}} $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} A(3,3)+\frac{4}{33} \sqrt{5} A(5,3) $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} B(3,3)+\frac{4}{33} \sqrt{5} B(5,3) $|$ 0 $| ^$ d_{3z^2-r^2} $|$ \frac{2}{33} \sqrt{10} B(5,3)-\frac{1}{3} \sqrt{\frac{10}{7}} B(3,3) $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ \frac{1}{231} \sqrt{10} \left(11 \sqrt{7} A(3,3)-14 A(5,3)\right) $| ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{5}{7}} B(3,3)+\frac{4}{33} \sqrt{5} B(5,3) $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ -\frac{1}{231} \sqrt{5} \left(11 \sqrt{7} A(3,3)+28 A(5,3)\right) $|$ 0 $| ^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ \frac{5}{33} \sqrt{2} B(5,3)-\frac{1}{3} \sqrt{\frac{2}{7}} B(3,3) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{2}{7}} A(3,3)-\frac{5}{33} \sqrt{2} A(5,3) $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 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]] | | | | | | ###