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| physics_chemistry:point_groups:cs:orientation_z [2018/03/24 23:08] – Maurits W. Haverkort | physics_chemistry:point_groups:cs:orientation_z [2018/03/30 10:01] – Maurits W. Haverkort |
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| $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ | $$V(r,\theta,\phi) = \sum_{k=0}^{\infty} \sum_{m=-k}^{k} A_{k,m}(r) C^{(m)}_k(\theta,\phi)$$ |
| Here $A_{k,m}(r)$ is a radial function and $C^{(m)}_k(\theta,\phi)$ a renormalised spherical harmonics. $$C^{(m)}_k(\theta,\phi)=\sqrt{\frac{4\pi}{2k+1}}Y^{(m)}_k(\theta,\phi)$$ | 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 ??? Point group with orientation ??? the form of the expansion coefficients is: | The presence of symmetry induces relations between the expansion coefficients such that $V(r,\theta,\phi)$ is invariant under all symmetry operations. For the Cs Point group with orientation Z the form of the expansion coefficients is: |
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| ### | ### |
| |
| ==== Input format suitable for Mathematica (Quanty.nb) ==== | ==== Expansion ==== |
| |
| ### | ### |
| $$A_{k,m} = \begin{cases} | $$A_{k,m} = \begin{cases} |
| A(0,0) & k=0\land m=0 \\ | A(0,0) & k=0\land m=0 \\ |
| -A(1,1)+i \text{Ap}(1,1) & k=1\land m=-1 \\ | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| A(1,1)+i \text{Ap}(1,1) & k=1\land m=1 \\ | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| A(2,2)-i \text{Ap}(2,2) & k=2\land m=-2 \\ | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| A(2,0) & k=2\land m=0 \\ | A(2,0) & k=2\land m=0 \\ |
| A(2,2)+i \text{Ap}(2,2) & k=2\land m=2 \\ | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| -A(3,3)+i \text{Ap}(3,3) & k=3\land m=-3 \\ | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| -A(3,1)+i \text{Ap}(3,1) & k=3\land m=-1 \\ | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| A(3,1)+i \text{Ap}(3,1) & k=3\land m=1 \\ | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| A(3,3)+i \text{Ap}(3,3) & k=3\land m=3 \\ | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| A(4,4)-i \text{Ap}(4,4) & k=4\land m=-4 \\ | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| A(4,2)-i \text{Ap}(4,2) & k=4\land m=-2 \\ | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| A(4,0) & k=4\land m=0 \\ | A(4,0) & k=4\land m=0 \\ |
| A(4,2)+i \text{Ap}(4,2) & k=4\land m=2 \\ | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| A(4,4)+i \text{Ap}(4,4) & k=4\land m=4 \\ | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| -A(5,5)+i \text{Ap}(5,5) & k=5\land m=-5 \\ | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| -A(5,3)+i \text{Ap}(5,3) & k=5\land m=-3 \\ | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| -A(5,1)+i \text{Ap}(5,1) & k=5\land m=-1 \\ | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| A(5,1)+i \text{Ap}(5,1) & k=5\land m=1 \\ | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| A(5,3)+i \text{Ap}(5,3) & k=5\land m=3 \\ | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| A(5,5)+i \text{Ap}(5,5) & k=5\land m=5 \\ | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| A(6,6)-i \text{Ap}(6,6) & k=6\land m=-6 \\ | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| A(6,4)-i \text{Ap}(6,4) & k=6\land m=-4 \\ | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| A(6,2)-i \text{Ap}(6,2) & k=6\land m=-2 \\ | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| A(6,0) & k=6\land m=0 \\ | A(6,0) & k=6\land m=0 \\ |
| A(6,2)+i \text{Ap}(6,2) & k=6\land m=2 \\ | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| A(6,4)+i \text{Ap}(6,4) & k=6\land m=4 \\ | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| A(6,6)+i \text{Ap}(6,6) & k=6\land m=6 | A(6,6)+i B(6,6) & k=6\land m=6 |
| \end{cases}$$ | \end{cases}$$ |
| | |
| | ### |
| | |
| | ==== Input format suitable for Mathematica (Quanty.nb) ==== |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| |
| ### | ### |
| |
| Akm = {{0, 0, A(0,0)} , | Akm = {{0, 0, A(0,0)} , |
| {1,-1, (-1)*(A(1,1)) + ((+1*I))*(Ap(1,1))} , | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| {1, 1, A(1,1) + ((+1*I))*(Ap(1,1))} , | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| {2, 0, A(2,0)} , | {2, 0, A(2,0)} , |
| {2,-2, A(2,2) + ((+-1*I))*(Ap(2,2))} , | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| {2, 2, A(2,2) + ((+1*I))*(Ap(2,2))} , | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| {3,-1, (-1)*(A(3,1)) + ((+1*I))*(Ap(3,1))} , | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| {3, 1, A(3,1) + ((+1*I))*(Ap(3,1))} , | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| {3,-3, (-1)*(A(3,3)) + ((+1*I))*(Ap(3,3))} , | {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} , |
| {3, 3, A(3,3) + ((+1*I))*(Ap(3,3))} , | {3, 3, A(3,3) + (I)*(B(3,3))} , |
| {4, 0, A(4,0)} , | {4, 0, A(4,0)} , |
| {4,-2, A(4,2) + ((+-1*I))*(Ap(4,2))} , | {4,-2, A(4,2) + (-I)*(B(4,2))} , |
| {4, 2, A(4,2) + ((+1*I))*(Ap(4,2))} , | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| {4,-4, A(4,4) + ((+-1*I))*(Ap(4,4))} , | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| {4, 4, A(4,4) + ((+1*I))*(Ap(4,4))} , | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| {5,-1, (-1)*(A(5,1)) + ((+1*I))*(Ap(5,1))} , | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| {5, 1, A(5,1) + ((+1*I))*(Ap(5,1))} , | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| {5,-3, (-1)*(A(5,3)) + ((+1*I))*(Ap(5,3))} , | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| {5, 3, A(5,3) + ((+1*I))*(Ap(5,3))} , | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| {5,-5, (-1)*(A(5,5)) + ((+1*I))*(Ap(5,5))} , | {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} , |
| {5, 5, A(5,5) + ((+1*I))*(Ap(5,5))} , | {5, 5, A(5,5) + (I)*(B(5,5))} , |
| {6, 0, A(6,0)} , | {6, 0, A(6,0)} , |
| {6,-2, A(6,2) + ((+-1*I))*(Ap(6,2))} , | {6,-2, A(6,2) + (-I)*(B(6,2))} , |
| {6, 2, A(6,2) + ((+1*I))*(Ap(6,2))} , | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| {6,-4, A(6,4) + ((+-1*I))*(Ap(6,4))} , | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| {6, 4, A(6,4) + ((+1*I))*(Ap(6,4))} , | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| {6,-6, A(6,6) + ((+-1*I))*(Ap(6,6))} , | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| {6, 6, A(6,6) + ((+1*I))*(Ap(6,6))} } | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| |
| </code> | </code> |
| The operator representing the potential in second quantisation is given as: | The operator representing the potential in second quantisation is given as: |
| $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ | $$ O = \sum_{n'',l'',m'',n',l',m'} \left\langle \psi_{n'',l'',m''}(r,\theta,\phi) \left| V(r,\theta,\phi) \right| \psi_{n',l',m'}(r,\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| For the quantisation of the wave-function we can choose a basis of spherical harmonics times some radial function $\psi_{n,l,m}(r,\theta,\phi)=R_{n,l}(r)Y_{m}^{(l)}(\theta,\phi)$. With this choice we can separate the radial part from the angular part for the evaluation of the operator. With the definition | 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. |
| $$ A_{l'',l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ | $$ A_{n''l'',n'l'}(k,m) = \left\langle R_{n'',l''} \left| A_{k,m}(r) \right| R_{n',l'} \right\rangle $$ |
| | Note the difference between the function $A_{k,m}$ and the parameter $A_{n''l'',n'l'}(k,m)$ |
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| | ### |
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| | ### |
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| we can express the operator as | we can express the operator as |
| $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{l'',l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ | $$ O = \sum_{n'',l'',m'',n',l',m',k,m} A_{n''l'',n'l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle a^{\dagger}_{n'',l'',m''}a^{\phantom{\dagger}}_{n',l',m'}$$ |
| The coefficient in front of the creation and annihilation operators | |
| $$ \sum_{k,m} A_{l'',l'}(k,m) \left\langle Y_{l''}^{(m'')}(\theta,\phi) \left| C_{k}^{(m)}(\theta,\phi) \right| Y_{l'}^{(m')}(\theta,\phi) \right\rangle $$ | |
| is shown in the table below. | |
| 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. | |
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| ### | ### |
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| | ### |
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| | 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. |
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| | ### |
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| ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== | ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== |
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| | ### |
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| | 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 |
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| | ### |
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| ### | ### |
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| ==== One particle coupling on a basis of symmetry adapted functions ==== | ==== One particle coupling on a basis of symmetry adapted functions ==== |
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| | ### |
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| | After rotation we find |
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| | ### |
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| ### | ### |
| ### | ### |
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| ===== Potential for s orbitals ===== | ===== Coupling for a single shell ===== |
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| ===== Potential for p orbitals ===== | |
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| ===== Potential for d orbitals ===== | |
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| ===== Potential for f orbitals ===== | ### |
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| ===== Potential for s-p orbital mixing ===== | 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 $l'$. |
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| ===== Potential for s-d orbital mixing ===== | ### |
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| ===== Potential for s-f orbital mixing ===== | |
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| ===== Potential for p-d orbital mixing ===== | |
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| ===== Potential for p-f orbital mixing ===== | ### |
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| ===== Potential for d-f orbital mixing ===== | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
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| | ### |
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| | ==== Potential for s orbitals ==== |
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| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
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| | ### |
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| | $$A_{k,m} = \begin{cases} |
| | \text{Eag} & k=0\land m=0 \\ |
| | 0 & \text{True} |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
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| | ### |
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| | <code Quanty Akm_Cs_Z.Quanty.nb> |
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| | Akm[k_,m_]:=Piecewise[{{Eag, k == 0 && m == 0}}, 0] |
| | |
| | </code> |
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| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
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| | ### |
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| | <code Quanty Akm_Cs_Z.Quanty> |
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| | Akm = {{0, 0, Eag} } |
| | |
| | </code> |
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| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
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| | ### |
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| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \text{Eag} $| |
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| | ### |
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| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
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| | ### |
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| | | $ $ ^ $ \text{s} $ ^ |
| | ^$ \text{s} $|$ \text{Eag} $| |
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| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
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| | ### |
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| | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ |
| | ^$ \text{s} $|$ 1 $| |
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| | ### |
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| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
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| | ### |
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| | ^ ^$$\text{Eag}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_0_1.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2 \sqrt{\pi }}$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{3} (\text{Epxpx}+\text{Epypy}+\text{Epzpz}) & k=0\land m=0 \\ |
| | \frac{5 (\text{Epxpx}+2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=-2 \\ |
| | \frac{5 (\text{Epzpx}+i \text{Epypz})}{\sqrt{6}} & k=2\land m=-1 \\ |
| | -\frac{5}{6} (\text{Epxpx}+\text{Epypy}-2 \text{Epzpz}) & k=2\land m=0 \\ |
| | \frac{5 i (\text{Epypz}+i \text{Epzpx})}{\sqrt{6}} & k=2\land m=1 \\ |
| | \frac{5 (\text{Epxpx}-2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=2 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Epxpx + Epypy + Epzpz)/3, k == 0 && m == 0}, {(5*(Epxpx + (2*I)*Epypx - Epypy))/(2*Sqrt[6]), k == 2 && m == -2}, {(5*(I*Epypz + Epzpx))/Sqrt[6], k == 2 && m == -1}, {(-5*(Epxpx + Epypy - 2*Epzpz))/6, k == 2 && m == 0}, {((5*I)*(Epypz + I*Epzpx))/Sqrt[6], k == 2 && m == 1}, {(5*(Epxpx - (2*I)*Epypx - Epypy))/(2*Sqrt[6]), k == 2 && m == 2}}, 0] |
| | |
| | </code> |
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| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
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| | ### |
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| | <code Quanty Akm_Cs_Z.Quanty> |
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| | Akm = {{0, 0, (1/3)*(Epxpx + Epypy + Epzpz)} , |
| | {2, 0, (-5/6)*(Epxpx + Epypy + (-2)*(Epzpz))} , |
| | {2,-1, (5)*((1/(sqrt(6)))*((I)*(Epypz) + Epzpx))} , |
| | {2, 1, (5*I)*((1/(sqrt(6)))*(Epypz + (I)*(Epzpx)))} , |
| | {2, 2, (5/2)*((1/(sqrt(6)))*(Epxpx + (-2*I)*(Epypx) + (-1)*(Epypy)))} , |
| | {2,-2, (5/2)*((1/(sqrt(6)))*(Epxpx + (2*I)*(Epypx) + (-1)*(Epypy)))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{\text{Epxpx}+\text{Epypy}}{2} $|$ \frac{\text{Epzpx}+i \text{Epypz}}{\sqrt{2}} $|$ \frac{1}{2} (-\text{Epxpx}-2 i \text{Epypx}+\text{Epypy}) $| |
| | ^$ {Y_{0}^{(1)}} $|$ \frac{\text{Epzpx}-i \text{Epypz}}{\sqrt{2}} $|$ \text{Epzpz} $|$ -\frac{\text{Epzpx}+i \text{Epypz}}{\sqrt{2}} $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{1}{2} (-\text{Epxpx}+2 i \text{Epypx}+\text{Epypy}) $|$ \frac{i (\text{Epypz}+i \text{Epzpx})}{\sqrt{2}} $|$ \frac{\text{Epxpx}+\text{Epypy}}{2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ |
| | ^$ p_x $|$ \text{Epxpx} $|$ \text{Epypx} $|$ \text{Epzpx} $| |
| | ^$ p_y $|$ \text{Epypx} $|$ \text{Epypy} $|$ \text{Epypz} $| |
| | ^$ p_z $|$ \text{Epzpx} $|$ \text{Epypz} $|$ \text{Epzpz} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| |
| | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| |
| | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Epxpx}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_1_1.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} x$$ | ::: | |
| | ^ ^$$\text{Epypy}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_1_2.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} y$$ | ::: | |
| | ^ ^$$\text{Epzpz}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_1_3.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{3}{\pi }} z$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | \frac{1}{5} (\text{Edx2y2dx2y2}+\text{Edxydxy}+\text{Edxzdxz}+\text{Edyzdyz}+\text{Edz2dz2}) & k=0\land m=0 \\ |
| | \frac{-4 i \text{Edxydz2}+\sqrt{3} \text{Edxzdxz}+2 i \sqrt{3} \text{Edyzdxz}-\sqrt{3} \text{Edyzdyz}-4 \text{Edz2dx2y2}}{2 \sqrt{2}} & k=2\land m=-2 \\ |
| | \frac{i \sqrt{3} \text{Edxydxz}+\sqrt{3} \text{Edxydyz}+\sqrt{3} \text{Edxzdx2y2}-i \sqrt{3} \text{Edyzdx2y2}+i \text{Edyzdz2}+\text{Edz2dxz}}{\sqrt{2}} & k=2\land m=-1 \\ |
| | -\text{Edx2y2dx2y2}-\text{Edxydxy}+\frac{\text{Edxzdxz}+\text{Edyzdyz}}{2}+\text{Edz2dz2} & k=2\land m=0 \\ |
| | \frac{i \left(\sqrt{3} \text{Edxydxz}+i \sqrt{3} \text{Edxydyz}+i \sqrt{3} \text{Edxzdx2y2}-\sqrt{3} \text{Edyzdx2y2}+\text{Edyzdz2}+i \text{Edz2dxz}\right)}{\sqrt{2}} & k=2\land m=1 \\ |
| | \frac{4 i \text{Edxydz2}+\sqrt{3} \text{Edxzdxz}-2 i \sqrt{3} \text{Edyzdxz}-\sqrt{3} \text{Edyzdyz}-4 \text{Edz2dx2y2}}{2 \sqrt{2}} & k=2\land m=2 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}+2 i \text{Edxydx2y2}-\text{Edxydxy}) & k=4\land m=-4 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{5}} (i \text{Edxydxz}-\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}) & k=4\land m=-3 \\ |
| | \frac{3 \left(i \sqrt{3} \text{Edxydz2}+\text{Edxzdxz}+2 i \text{Edyzdxz}-\text{Edyzdyz}+\sqrt{3} \text{Edz2dx2y2}\right)}{\sqrt{10}} & k=4\land m=-2 \\ |
| | \frac{3 \left(-i \text{Edxydxz}-\text{Edxydyz}-\text{Edxzdx2y2}+i \text{Edyzdx2y2}+2 i \sqrt{3} \text{Edyzdz2}+2 \sqrt{3} \text{Edz2dxz}\right)}{2 \sqrt{5}} & k=4\land m=-1 \\ |
| | \frac{3}{10} (\text{Edx2y2dx2y2}+\text{Edxydxy}-4 (\text{Edxzdxz}+\text{Edyzdyz})+6 \text{Edz2dz2}) & k=4\land m=0 \\ |
| | \frac{3 \left(-i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}+2 i \sqrt{3} \text{Edyzdz2}-2 \sqrt{3} \text{Edz2dxz}\right)}{2 \sqrt{5}} & k=4\land m=1 \\ |
| | \frac{3 \left(-i \sqrt{3} \text{Edxydz2}+\text{Edxzdxz}-2 i \text{Edyzdxz}-\text{Edyzdyz}+\sqrt{3} \text{Edz2dx2y2}\right)}{\sqrt{10}} & k=4\land m=2 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{5}} (i \text{Edxydxz}+\text{Edxydyz}-\text{Edxzdx2y2}+i \text{Edyzdx2y2}) & k=4\land m=3 \\ |
| | \frac{3}{2} \sqrt{\frac{7}{10}} (\text{Edx2y2dx2y2}-2 i \text{Edxydx2y2}-\text{Edxydxy}) & k=4\land m=4 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)/5, k == 0 && m == 0}, {((-4*I)*Edxydz2 + Sqrt[3]*Edxzdxz + (2*I)*Sqrt[3]*Edyzdxz - Sqrt[3]*Edyzdyz - 4*Edz2dx2y2)/(2*Sqrt[2]), k == 2 && m == -2}, {(I*Sqrt[3]*Edxydxz + Sqrt[3]*Edxydyz + Sqrt[3]*Edxzdx2y2 - I*Sqrt[3]*Edyzdx2y2 + I*Edyzdz2 + Edz2dxz)/Sqrt[2], k == 2 && m == -1}, {-Edx2y2dx2y2 - Edxydxy + (Edxzdxz + Edyzdyz)/2 + Edz2dz2, k == 2 && m == 0}, {(I*(Sqrt[3]*Edxydxz + I*Sqrt[3]*Edxydyz + I*Sqrt[3]*Edxzdx2y2 - Sqrt[3]*Edyzdx2y2 + Edyzdz2 + I*Edz2dxz))/Sqrt[2], k == 2 && m == 1}, {((4*I)*Edxydz2 + Sqrt[3]*Edxzdxz - (2*I)*Sqrt[3]*Edyzdxz - Sqrt[3]*Edyzdyz - 4*Edz2dx2y2)/(2*Sqrt[2]), k == 2 && m == 2}, {(3*Sqrt[7/10]*(Edx2y2dx2y2 + (2*I)*Edxydx2y2 - Edxydxy))/2, k == 4 && m == -4}, {(3*Sqrt[7/5]*(I*Edxydxz - Edxydyz + Edxzdx2y2 + I*Edyzdx2y2))/2, k == 4 && m == -3}, {(3*(I*Sqrt[3]*Edxydz2 + Edxzdxz + (2*I)*Edyzdxz - Edyzdyz + Sqrt[3]*Edz2dx2y2))/Sqrt[10], k == 4 && m == -2}, {(3*((-I)*Edxydxz - Edxydyz - Edxzdx2y2 + I*Edyzdx2y2 + (2*I)*Sqrt[3]*Edyzdz2 + 2*Sqrt[3]*Edz2dxz))/(2*Sqrt[5]), k == 4 && m == -1}, {(3*(Edx2y2dx2y2 + Edxydxy - 4*(Edxzdxz + Edyzdyz) + 6*Edz2dz2))/10, k == 4 && m == 0}, {(3*((-I)*Edxydxz + Edxydyz + Edxzdx2y2 + I*Edyzdx2y2 + (2*I)*Sqrt[3]*Edyzdz2 - 2*Sqrt[3]*Edz2dxz))/(2*Sqrt[5]), k == 4 && m == 1}, {(3*((-I)*Sqrt[3]*Edxydz2 + Edxzdxz - (2*I)*Edyzdxz - Edyzdyz + Sqrt[3]*Edz2dx2y2))/Sqrt[10], k == 4 && m == 2}, {(3*Sqrt[7/5]*(I*Edxydxz + Edxydyz - Edxzdx2y2 + I*Edyzdx2y2))/2, k == 4 && m == 3}, {(3*Sqrt[7/10]*(Edx2y2dx2y2 - (2*I)*Edxydx2y2 - Edxydxy))/2, k == 4 && m == 4}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, (1/5)*(Edx2y2dx2y2 + Edxydxy + Edxzdxz + Edyzdyz + Edz2dz2)} , |
| | {2, 0, (-1)*(Edx2y2dx2y2) + (-1)*(Edxydxy) + (1/2)*(Edxzdxz + Edyzdyz) + Edz2dz2} , |
| | {2,-1, (1/(sqrt(2)))*((I)*((sqrt(3))*(Edxydxz)) + (sqrt(3))*(Edxydyz) + (sqrt(3))*(Edxzdx2y2) + (-I)*((sqrt(3))*(Edyzdx2y2)) + (I)*(Edyzdz2) + Edz2dxz)} , |
| | {2, 1, (I)*((1/(sqrt(2)))*((sqrt(3))*(Edxydxz) + (I)*((sqrt(3))*(Edxydyz)) + (I)*((sqrt(3))*(Edxzdx2y2)) + (-1)*((sqrt(3))*(Edyzdx2y2)) + Edyzdz2 + (I)*(Edz2dxz)))} , |
| | {2,-2, (1/2)*((1/(sqrt(2)))*((-4*I)*(Edxydz2) + (sqrt(3))*(Edxzdxz) + (2*I)*((sqrt(3))*(Edyzdxz)) + (-1)*((sqrt(3))*(Edyzdyz)) + (-4)*(Edz2dx2y2)))} , |
| | {2, 2, (1/2)*((1/(sqrt(2)))*((4*I)*(Edxydz2) + (sqrt(3))*(Edxzdxz) + (-2*I)*((sqrt(3))*(Edyzdxz)) + (-1)*((sqrt(3))*(Edyzdyz)) + (-4)*(Edz2dx2y2)))} , |
| | {4, 0, (3/10)*(Edx2y2dx2y2 + Edxydxy + (-4)*(Edxzdxz + Edyzdyz) + (6)*(Edz2dz2))} , |
| | {4,-1, (3/2)*((1/(sqrt(5)))*((-I)*(Edxydxz) + (-1)*(Edxydyz) + (-1)*(Edxzdx2y2) + (I)*(Edyzdx2y2) + (2*I)*((sqrt(3))*(Edyzdz2)) + (2)*((sqrt(3))*(Edz2dxz))))} , |
| | {4, 1, (3/2)*((1/(sqrt(5)))*((-I)*(Edxydxz) + Edxydyz + Edxzdx2y2 + (I)*(Edyzdx2y2) + (2*I)*((sqrt(3))*(Edyzdz2)) + (-2)*((sqrt(3))*(Edz2dxz))))} , |
| | {4, 2, (3)*((1/(sqrt(10)))*((-I)*((sqrt(3))*(Edxydz2)) + Edxzdxz + (-2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (sqrt(3))*(Edz2dx2y2)))} , |
| | {4,-2, (3)*((1/(sqrt(10)))*((I)*((sqrt(3))*(Edxydz2)) + Edxzdxz + (2*I)*(Edyzdxz) + (-1)*(Edyzdyz) + (sqrt(3))*(Edz2dx2y2)))} , |
| | {4,-3, (3/2)*((sqrt(7/5))*((I)*(Edxydxz) + (-1)*(Edxydyz) + Edxzdx2y2 + (I)*(Edyzdx2y2)))} , |
| | {4, 3, (3/2)*((sqrt(7/5))*((I)*(Edxydxz) + Edxydyz + (-1)*(Edxzdx2y2) + (I)*(Edyzdx2y2)))} , |
| | {4, 4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (-2*I)*(Edxydx2y2) + (-1)*(Edxydxy)))} , |
| | {4,-4, (3/2)*((sqrt(7/10))*(Edx2y2dx2y2 + (2*I)*(Edxydx2y2) + (-1)*(Edxydxy)))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ {Y_{-2}^{(2)}} $|$ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $|$ \frac{1}{2} (i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}-i \text{Edyzdx2y2}) $|$ \frac{\text{Edz2dx2y2}+i \text{Edxydz2}}{\sqrt{2}} $|$ -\frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}-\text{Edxzdx2y2})+\text{Edyzdx2y2}) $|$ \frac{1}{2} (\text{Edx2y2dx2y2}+2 i \text{Edxydx2y2}-\text{Edxydxy}) $| |
| | ^$ {Y_{-1}^{(2)}} $|$ \frac{1}{2} (-i \text{Edxydxz}+\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2}) $|$ \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} $|$ \frac{\text{Edz2dxz}+i \text{Edyzdz2}}{\sqrt{2}} $|$ \frac{1}{2} (-\text{Edxzdxz}-2 i \text{Edyzdxz}+\text{Edyzdyz}) $|$ \frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}-\text{Edxzdx2y2})+\text{Edyzdx2y2}) $| |
| | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Edz2dx2y2}-i \text{Edxydz2}}{\sqrt{2}} $|$ \frac{\text{Edz2dxz}-i \text{Edyzdz2}}{\sqrt{2}} $|$ \text{Edz2dz2} $|$ \frac{-\text{Edz2dxz}-i \text{Edyzdz2}}{\sqrt{2}} $|$ \frac{\text{Edz2dx2y2}+i \text{Edxydz2}}{\sqrt{2}} $| |
| | ^$ {Y_{1}^{(2)}} $|$ \frac{1}{2} i (\text{Edxydxz}-i \text{Edxydyz}+i \text{Edxzdx2y2}+\text{Edyzdx2y2}) $|$ \frac{1}{2} (-\text{Edxzdxz}+2 i \text{Edyzdxz}+\text{Edyzdyz}) $|$ \frac{i (\text{Edyzdz2}+i \text{Edz2dxz})}{\sqrt{2}} $|$ \frac{\text{Edxzdxz}+\text{Edyzdyz}}{2} $|$ -\frac{1}{2} i (\text{Edxydxz}-i (\text{Edxydyz}+\text{Edxzdx2y2})-\text{Edyzdx2y2}) $| |
| | ^$ {Y_{2}^{(2)}} $|$ \frac{1}{2} (\text{Edx2y2dx2y2}-2 i \text{Edxydx2y2}-\text{Edxydxy}) $|$ -\frac{1}{2} i (\text{Edxydxz}-i \text{Edxydyz}+i \text{Edxzdx2y2}+\text{Edyzdx2y2}) $|$ \frac{\text{Edz2dx2y2}-i \text{Edxydz2}}{\sqrt{2}} $|$ \frac{1}{2} i (\text{Edxydxz}+i (\text{Edxydyz}+\text{Edxzdx2y2}+i \text{Edyzdx2y2})) $|$ \frac{\text{Edx2y2dx2y2}+\text{Edxydxy}}{2} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ d_{x^2-y^2} $|$ \text{Edx2y2dx2y2} $|$ \text{Edz2dx2y2} $|$ \text{Edyzdx2y2} $|$ \text{Edxzdx2y2} $|$ \text{Edxydx2y2} $| |
| | ^$ d_{3z^2-r^2} $|$ \text{Edz2dx2y2} $|$ \text{Edz2dz2} $|$ \text{Edyzdz2} $|$ \text{Edz2dxz} $|$ \text{Edxydz2} $| |
| | ^$ d_{\text{yz}} $|$ \text{Edyzdx2y2} $|$ \text{Edyzdz2} $|$ \text{Edyzdyz} $|$ \text{Edyzdxz} $|$ \text{Edxydyz} $| |
| | ^$ d_{\text{xz}} $|$ \text{Edxzdx2y2} $|$ \text{Edz2dxz} $|$ \text{Edyzdxz} $|$ \text{Edxzdxz} $|$ \text{Edxydxz} $| |
| | ^$ d_{\text{xy}} $|$ \text{Edxydx2y2} $|$ \text{Edxydz2} $|$ \text{Edxydyz} $|$ \text{Edxydxz} $|$ \text{Edxydxy} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| |
| | ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| |
| | ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\text{Edx2y2dx2y2}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_2_1.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$$ | ::: | |
| | ^ ^$$\text{Edz2dz2}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_2_2.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$$ | ::: | |
| | ^ ^$$\text{Edyzdyz}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_2_3.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$$ | ::: | |
| | ^ ^$$\text{Edxzdxz}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_2_4.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$$ | ::: | |
| | ^ ^$$\text{Edxydxy}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_2_5.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for f orbitals ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{-3}^{(3)}} $|$ A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)-i B(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i B(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i B(6,2)) $|$ 0 $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i B(4,4))-5 \sqrt{5} (A(6,4)-i B(6,4))\right) $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)-i B(6,6)) $| |
| | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) $|$ 0 $|$ -\frac{2 (A(2,2)-i B(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i B(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i B(6,2)) $|$ 0 $|$ \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)-i B(4,4))+30 (A(6,4)-i B(6,4))\right) $|$ 0 $| |
| | ^$ {Y_{-1}^{(3)}} $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i B(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i B(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i B(6,2)) $|$ 0 $|$ A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0) $|$ 0 $|$ \frac{2 \left(-143 \sqrt{6} (A(2,2)-i B(2,2))-65 \sqrt{10} (A(4,2)-i B(4,2))-25 \sqrt{105} (A(6,2)-i B(6,2))\right)}{2145} $|$ 0 $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i B(4,4))-5 \sqrt{5} (A(6,4)-i B(6,4))\right) $| |
| | ^$ {Y_{0}^{(3)}} $|$ 0 $|$ -\frac{2 (A(2,2)+i B(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i B(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i B(6,2)) $|$ 0 $|$ A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0)) $|$ 0 $|$ -\frac{2 (A(2,2)-i B(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i B(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i B(6,2)) $|$ 0 $| |
| | ^$ {Y_{1}^{(3)}} $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i B(4,4))-5 \sqrt{5} (A(6,4)+i B(6,4))\right) $|$ 0 $|$ \frac{2 \left(-143 \sqrt{6} (A(2,2)+i B(2,2))-65 \sqrt{10} (A(4,2)+i B(4,2))-25 \sqrt{105} (A(6,2)+i B(6,2))\right)}{2145} $|$ 0 $|$ A(0,0)+\frac{1}{5} A(2,0)+\frac{1}{33} A(4,0)-\frac{25}{143} A(6,0) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)-i B(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i B(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i B(6,2)) $| |
| | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)+i B(4,4))+30 (A(6,4)+i B(6,4))\right) $|$ 0 $|$ -\frac{2 (A(2,2)+i B(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i B(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i B(6,2)) $|$ 0 $|$ A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) $|$ 0 $| |
| | ^$ {Y_{3}^{(3)}} $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)+i B(6,6)) $|$ 0 $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i B(4,4))-5 \sqrt{5} (A(6,4)+i B(6,4))\right) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i B(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i B(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i B(6,2)) $|$ 0 $|$ A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ f_{\text{xyz}} $|$ \frac{1}{429} \left(429 A(0,0)-91 A(4,0)-13 \sqrt{70} A(4,4)+30 A(6,0)-30 \sqrt{14} A(6,4)\right) $|$ 0 $|$ 0 $|$ \frac{286 \sqrt{10} B(2,2)+65 \sqrt{6} B(4,2)-200 \sqrt{7} B(6,2)}{2145} $|$ 0 $|$ 0 $|$ -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} B(4,4)+30 B(6,4)\right) $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \frac{8580 A(0,0)-1144 A(2,0)+1144 \sqrt{6} A(2,2)+585 A(4,0)-390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)+125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)+125 \sqrt{231} A(6,6)}{8580} $|$ \frac{286 \sqrt{6} B(2,2)-780 \sqrt{10} B(4,2)-25 \sqrt{105} B(6,2)+125 \sqrt{231} B(6,6)}{8580} $|$ 0 $|$ \frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)-26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)-85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)+15 \sqrt{385} A(6,6)\right)}{8580} $|$ \frac{286 \sqrt{10} B(2,2)+5 \left(104 \sqrt{6} B(4,2)+\sqrt{7} \left(52 \sqrt{6} B(4,4)+65 B(6,2)-20 \sqrt{30} B(6,4)+15 \sqrt{55} B(6,6)\right)\right)}{8580} $|$ 0 $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ \frac{286 \sqrt{6} B(2,2)-780 \sqrt{10} B(4,2)-25 \sqrt{105} B(6,2)+125 \sqrt{231} B(6,6)}{8580} $|$ \frac{8580 A(0,0)-1144 A(2,0)-1144 \sqrt{6} A(2,2)+585 A(4,0)+390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)-125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)-125 \sqrt{231} A(6,6)}{8580} $|$ 0 $|$ \frac{-286 \sqrt{10} B(2,2)-5 \left(104 \sqrt{6} B(4,2)+\sqrt{7} \left(-52 \sqrt{6} B(4,4)+65 B(6,2)+20 \sqrt{30} B(6,4)+15 \sqrt{55} B(6,6)\right)\right)}{8580} $|$ \frac{-572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)+5 \left(13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{8580} $|$ 0 $| |
| | ^$ f_{x\left(5z^2-r^2\right)} $|$ \frac{286 \sqrt{10} B(2,2)+65 \sqrt{6} B(4,2)-200 \sqrt{7} B(6,2)}{2145} $|$ 0 $|$ 0 $|$ A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0)) $|$ 0 $|$ 0 $|$ \frac{-286 \sqrt{10} A(2,2)-65 \sqrt{6} A(4,2)+200 \sqrt{7} A(6,2)}{2145} $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ \frac{572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)-5 \left(13 \sqrt{15} A(4,0)-26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)-85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)+15 \sqrt{385} A(6,6)\right)}{8580} $|$ \frac{-286 \sqrt{10} B(2,2)-5 \left(104 \sqrt{6} B(4,2)+\sqrt{7} \left(-52 \sqrt{6} B(4,4)+65 B(6,2)+20 \sqrt{30} B(6,4)+15 \sqrt{55} B(6,6)\right)\right)}{8580} $|$ 0 $|$ \frac{1716 A(0,0)+91 A(4,0)+182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)+15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)+15 \sqrt{231} A(6,6)}{1716} $|$ \frac{286 \sqrt{6} B(2,2)-52 \sqrt{10} B(4,2)+35 \sqrt{105} B(6,2)-15 \sqrt{231} B(6,6)}{1716} $|$ 0 $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ \frac{286 \sqrt{10} B(2,2)+5 \left(104 \sqrt{6} B(4,2)+\sqrt{7} \left(52 \sqrt{6} B(4,4)+65 B(6,2)-20 \sqrt{30} B(6,4)+15 \sqrt{55} B(6,6)\right)\right)}{8580} $|$ \frac{-572 \sqrt{15} A(2,0)+572 \sqrt{10} A(2,2)+5 \left(13 \sqrt{15} A(4,0)+26 \sqrt{6} A(4,2)-13 \sqrt{42} A(4,4)+35 \sqrt{15} A(6,0)+85 \sqrt{7} A(6,2)+5 \sqrt{210} A(6,4)-15 \sqrt{385} A(6,6)\right)}{8580} $|$ 0 $|$ \frac{286 \sqrt{6} B(2,2)-52 \sqrt{10} B(4,2)+35 \sqrt{105} B(6,2)-15 \sqrt{231} B(6,6)}{1716} $|$ \frac{1716 A(0,0)+91 A(4,0)-182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)-15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)-15 \sqrt{231} A(6,6)}{1716} $|$ 0 $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$ -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} B(4,4)+30 B(6,4)\right) $|$ 0 $|$ 0 $|$ \frac{-286 \sqrt{10} A(2,2)-65 \sqrt{6} A(4,2)+200 \sqrt{7} A(6,2)}{2145} $|$ 0 $|$ 0 $|$ \frac{1}{429} \left(429 A(0,0)-91 A(4,0)+13 \sqrt{70} A(4,4)+30 A(6,0)+30 \sqrt{14} A(6,4)\right) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Rotation matrix used** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| |
| | ^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| |
| | ^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| |
| | ^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| |
| | ^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| |
| | ^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| |
| | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Irriducible representations and their onsite energy** > |
| | |
| | ### |
| | |
| | ^ ^$$\frac{1}{429} \left(429 A(0,0)-91 A(4,0)-13 \sqrt{70} A(4,4)+30 A(6,0)-30 \sqrt{14} A(6,4)\right)$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_1.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$$ | ::: | |
| | ^ ^$$\frac{8580 A(0,0)-1144 A(2,0)+1144 \sqrt{6} A(2,2)+585 A(4,0)-390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)+125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)+125 \sqrt{231} A(6,6)}{8580}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_2.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \cos (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )-5 \cos (2 \theta )-7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} x \left(5 x^2-15 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$\frac{8580 A(0,0)-1144 A(2,0)-1144 \sqrt{6} A(2,2)+585 A(4,0)+390 \sqrt{10} A(4,2)+195 \sqrt{70} A(4,4)-625 A(6,0)-125 \sqrt{105} A(6,2)-375 \sqrt{14} A(6,4)-125 \sqrt{231} A(6,6)}{8580}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_3.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{7}{\pi }} \sin (\theta ) \sin (\phi ) \left(10 \sin ^2(\theta ) \cos (2 \phi )+5 \cos (2 \theta )+7\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} y \left(-15 x^2+5 y^2-15 z^2+3\right)$$ | ::: | |
| | ^ ^$$A(0,0)+\frac{4}{15} A(2,0)+\frac{2}{429} (39 A(4,0)+50 A(6,0))$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_4.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$$ | ::: | |
| | ^ ^$$\frac{1716 A(0,0)+91 A(4,0)+182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)+15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)+15 \sqrt{231} A(6,6)}{1716}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_5.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} \sin (\theta ) \cos (\phi ) \left(2 \sin ^2(\theta ) \cos (2 \phi )+3 \cos (2 \theta )+1\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$-\frac{1}{16} \sqrt{\frac{105}{\pi }} x \left(x^2-3 y^2+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\frac{1716 A(0,0)+91 A(4,0)-182 \sqrt{10} A(4,2)-39 \sqrt{70} A(4,4)-195 A(6,0)-15 \sqrt{105} A(6,2)+75 \sqrt{14} A(6,4)-15 \sqrt{231} A(6,6)}{1716}$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_6.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{32} \sqrt{\frac{105}{\pi }} \sin (\theta ) \sin (\phi ) \left(-4 \sin ^2(\theta ) \cos (2 \phi )+6 \cos (2 \theta )+2\right)$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{16} \sqrt{\frac{105}{\pi }} y \left(-3 x^2+y^2+5 z^2-1\right)$$ | ::: | |
| | ^ ^$$\frac{1}{429} \left(429 A(0,0)-91 A(4,0)+13 \sqrt{70} A(4,4)+30 A(6,0)+30 \sqrt{14} A(6,4)\right)$$ | {{:physics_chemistry:pointgroup:cs_z_orb_3_7.png?50}} | |
| | |$$\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$$ | ::: | |
| | |$$\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$$ |$$\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$$ | ::: | |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ===== Coupling between two shells ===== |
| | |
| | |
| | |
| | ### |
| | |
| | Click on one of the subsections to expand it or <hiddenSwitch expand all> |
| | |
| | ### |
| | |
| | ==== Potential for s-p orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ -\frac{A(1,1)+i B(1,1)}{\sqrt{3}} $|$ 0 $|$ \frac{A(1,1)-i B(1,1)}{\sqrt{3}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ |
| | ^$ \text{s} $|$ -\sqrt{\frac{2}{3}} A(1,1) $|$ \sqrt{\frac{2}{3}} B(1,1) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for s-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ {Y_{0}^{(0)}} $|$ \frac{A(2,2)+i B(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,2)-i B(2,2)}{\sqrt{5}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ \text{s} $|$ \sqrt{\frac{2}{5}} A(2,2) $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ 0 $|$ -\sqrt{\frac{2}{5}} B(2,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for s-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {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 $|$ -\frac{A(3,1)+i B(3,1)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,1)-i B(3,1)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,3)-i B(3,3)}{\sqrt{7}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ \text{s} $|$ 0 $|$ \frac{1}{14} \left(\sqrt{21} A(3,1)-\sqrt{35} A(3,3)\right) $|$ -\frac{\sqrt{3} B(3,1)+\sqrt{5} B(3,3)}{2 \sqrt{7}} $|$ 0 $|$ \frac{\sqrt{5} A(3,1)+\sqrt{3} A(3,3)}{2 \sqrt{7}} $|$ \frac{1}{14} \left(\sqrt{35} B(3,1)-\sqrt{21} B(3,3)\right) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-d orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)+i B(3,1))-\sqrt{\frac{2}{5}} (A(1,1)+i B(1,1)) $|$ 0 $|$ \frac{1}{105} \left(7 \sqrt{15} (A(1,1)-i B(1,1))-9 \sqrt{10} (A(3,1)-i B(3,1))\right) $|$ 0 $|$ -\frac{3}{7} (A(3,3)-i B(3,3)) $| |
| | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ -\frac{A(1,1)+i B(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (A(3,1)+i B(3,1)) $|$ 0 $|$ \frac{7 A(1,1)+2 \sqrt{6} A(3,1)-i \left(7 B(1,1)+2 \sqrt{6} B(3,1)\right)}{7 \sqrt{5}} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$ \frac{3}{7} (A(3,3)+i B(3,3)) $|$ 0 $|$ \frac{3}{7} \sqrt{\frac{2}{5}} (A(3,1)+i B(3,1))-\frac{A(1,1)+i B(1,1)}{\sqrt{15}} $|$ 0 $|$ \sqrt{\frac{2}{5}} (A(1,1)-i B(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)-i B(3,1)) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ |
| | ^$ p_x $|$ \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)-15 A(3,3)\right) $|$ \sqrt{\frac{2}{15}} A(1,1)-\frac{6 A(3,1)}{7 \sqrt{5}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{5}} B(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} B(3,1)+\frac{3}{7} B(3,3) $| |
| | ^$ p_y $|$ \frac{1}{35} \left(-7 \sqrt{10} B(1,1)+\sqrt{15} B(3,1)+15 B(3,3)\right) $|$ \frac{6 B(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} B(1,1) $|$ 0 $|$ 0 $|$ \frac{1}{35} \left(-7 \sqrt{10} A(1,1)+\sqrt{15} A(3,1)+15 A(3,3)\right) $| |
| | ^$ p_z $|$ 0 $|$ 0 $|$ \sqrt{\frac{2}{5}} B(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} B(3,1) $|$ -\frac{1}{7} \sqrt{\frac{2}{5}} \left(7 A(1,1)+2 \sqrt{6} A(3,1)\right) $|$ 0 $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for p-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{-1}^{(1)}} $|$ \frac{3 (A(2,2)+i B(2,2))}{\sqrt{35}}-\frac{A(4,2)+i B(4,2)}{3 \sqrt{21}} $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)-i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i B(4,2)) $|$ 0 $|$ -\frac{2 (A(4,4)-i B(4,4))}{3 \sqrt{3}} $| |
| | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \sqrt{\frac{3}{35}} (A(2,2)+i B(2,2))+\frac{2 (A(4,2)+i B(4,2))}{3 \sqrt{7}} $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{35}} (A(2,2)-i B(2,2))+\frac{2 (A(4,2)-i B(4,2))}{3 \sqrt{7}} $|$ 0 $| |
| | ^$ {Y_{1}^{(1)}} $|$ -\frac{2 (A(4,4)+i B(4,4))}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i B(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i B(4,2)) $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ \frac{3 (A(2,2)-i B(2,2))}{\sqrt{35}}-\frac{A(4,2)-i B(4,2)}{3 \sqrt{21}} $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ p_x $|$ 0 $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)+81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)-2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)+7 B(4,4)\right)\right) $|$ 0 $|$ \frac{1}{210} \left(-9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)+5 \left(\sqrt{35} A(4,0)-2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ \sqrt{\frac{6}{35}} B(2,2)-\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} $|$ 0 $| |
| | ^$ p_y $|$ 0 $|$ \frac{1}{630} \left(54 \sqrt{14} B(2,2)+5 \sqrt{30} \left(\sqrt{7} B(4,2)-7 B(4,4)\right)\right) $|$ \frac{1}{630} \left(-27 \sqrt{21} A(2,0)-81 \sqrt{14} A(2,2)+5 \left(3 \sqrt{21} A(4,0)+2 \sqrt{210} A(4,2)+7 \sqrt{30} A(4,4)\right)\right) $|$ 0 $|$ -\sqrt{\frac{6}{35}} B(2,2)+\frac{B(4,2)}{\sqrt{14}}+\frac{B(4,4)}{3 \sqrt{2}} $|$ \frac{1}{210} \left(9 \sqrt{35} A(2,0)-3 \sqrt{210} A(2,2)-5 \left(\sqrt{35} A(4,0)+2 \sqrt{14} A(4,2)-7 \sqrt{2} A(4,4)\right)\right) $|$ 0 $| |
| | ^$ p_z $|$ -\sqrt{\frac{6}{35}} B(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} B(4,2) $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} A(2,2)+\frac{2}{3} \sqrt{\frac{2}{7}} A(4,2) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | ==== Potential for d-f orbital mixing ==== |
| | |
| | <hidden **Potential parameterized with onsite energies of irriducible representations** > |
| | |
| | ### |
| | |
| | $$A_{k,m} = \begin{cases} |
| | A(0,0) & k=0\land m=0 \\ |
| | -A(1,1)+i B(1,1) & k=1\land m=-1 \\ |
| | A(1,1)+i B(1,1) & k=1\land m=1 \\ |
| | A(2,2)-i B(2,2) & k=2\land m=-2 \\ |
| | A(2,0) & k=2\land m=0 \\ |
| | A(2,2)+i B(2,2) & k=2\land m=2 \\ |
| | -A(3,3)+i B(3,3) & k=3\land m=-3 \\ |
| | -A(3,1)+i B(3,1) & k=3\land m=-1 \\ |
| | A(3,1)+i B(3,1) & k=3\land m=1 \\ |
| | A(3,3)+i B(3,3) & k=3\land m=3 \\ |
| | A(4,4)-i B(4,4) & k=4\land m=-4 \\ |
| | A(4,2)-i B(4,2) & k=4\land m=-2 \\ |
| | A(4,0) & k=4\land m=0 \\ |
| | A(4,2)+i B(4,2) & k=4\land m=2 \\ |
| | A(4,4)+i B(4,4) & k=4\land m=4 \\ |
| | -A(5,5)+i B(5,5) & k=5\land m=-5 \\ |
| | -A(5,3)+i B(5,3) & k=5\land m=-3 \\ |
| | -A(5,1)+i B(5,1) & k=5\land m=-1 \\ |
| | A(5,1)+i B(5,1) & k=5\land m=1 \\ |
| | A(5,3)+i B(5,3) & k=5\land m=3 \\ |
| | A(5,5)+i B(5,5) & k=5\land m=5 \\ |
| | A(6,6)-i B(6,6) & k=6\land m=-6 \\ |
| | A(6,4)-i B(6,4) & k=6\land m=-4 \\ |
| | A(6,2)-i B(6,2) & k=6\land m=-2 \\ |
| | A(6,0) & k=6\land m=0 \\ |
| | A(6,2)+i B(6,2) & k=6\land m=2 \\ |
| | A(6,4)+i B(6,4) & k=6\land m=4 \\ |
| | A(6,6)+i B(6,6) & k=6\land m=6 |
| | \end{cases}$$ |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **Input format suitable for Mathematica (Quanty.nb)** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty.nb> |
| | |
| | Akm[k_,m_]:=Piecewise[{{A[0, 0], k == 0 && m == 0}, {-A[1, 1] + I*B[1, 1], k == 1 && m == -1}, {A[1, 1] + I*B[1, 1], k == 1 && m == 1}, {A[2, 2] - I*B[2, 2], k == 2 && m == -2}, {A[2, 0], k == 2 && m == 0}, {A[2, 2] + I*B[2, 2], k == 2 && m == 2}, {-A[3, 3] + I*B[3, 3], k == 3 && m == -3}, {-A[3, 1] + I*B[3, 1], k == 3 && m == -1}, {A[3, 1] + I*B[3, 1], k == 3 && m == 1}, {A[3, 3] + I*B[3, 3], k == 3 && m == 3}, {A[4, 4] - I*B[4, 4], k == 4 && m == -4}, {A[4, 2] - I*B[4, 2], k == 4 && m == -2}, {A[4, 0], k == 4 && m == 0}, {A[4, 2] + I*B[4, 2], k == 4 && m == 2}, {A[4, 4] + I*B[4, 4], k == 4 && m == 4}, {-A[5, 5] + I*B[5, 5], k == 5 && m == -5}, {-A[5, 3] + I*B[5, 3], k == 5 && m == -3}, {-A[5, 1] + I*B[5, 1], k == 5 && m == -1}, {A[5, 1] + I*B[5, 1], k == 5 && m == 1}, {A[5, 3] + I*B[5, 3], k == 5 && m == 3}, {A[5, 5] + I*B[5, 5], k == 5 && m == 5}, {A[6, 6] - I*B[6, 6], k == 6 && m == -6}, {A[6, 4] - I*B[6, 4], k == 6 && m == -4}, {A[6, 2] - I*B[6, 2], k == 6 && m == -2}, {A[6, 0], k == 6 && m == 0}, {A[6, 2] + I*B[6, 2], k == 6 && m == 2}, {A[6, 4] + I*B[6, 4], k == 6 && m == 4}, {A[6, 6] + I*B[6, 6], k == 6 && m == 6}}, 0] |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden><hidden **Input format suitable for Quanty** > |
| | |
| | ### |
| | |
| | <code Quanty Akm_Cs_Z.Quanty> |
| | |
| | Akm = {{0, 0, A(0,0)} , |
| | {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} , |
| | {1, 1, A(1,1) + (I)*(B(1,1))} , |
| | {2, 0, A(2,0)} , |
| | {2,-2, A(2,2) + (-I)*(B(2,2))} , |
| | {2, 2, A(2,2) + (I)*(B(2,2))} , |
| | {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} , |
| | {3, 1, A(3,1) + (I)*(B(3,1))} , |
| | {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,-2, A(4,2) + (-I)*(B(4,2))} , |
| | {4, 2, A(4,2) + (I)*(B(4,2))} , |
| | {4,-4, A(4,4) + (-I)*(B(4,4))} , |
| | {4, 4, A(4,4) + (I)*(B(4,4))} , |
| | {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} , |
| | {5, 1, A(5,1) + (I)*(B(5,1))} , |
| | {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} , |
| | {5, 3, A(5,3) + (I)*(B(5,3))} , |
| | {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,-2, A(6,2) + (-I)*(B(6,2))} , |
| | {6, 2, A(6,2) + (I)*(B(6,2))} , |
| | {6,-4, A(6,4) + (-I)*(B(6,4))} , |
| | {6, 4, A(6,4) + (I)*(B(6,4))} , |
| | {6,-6, A(6,6) + (-I)*(B(6,6))} , |
| | {6, 6, A(6,6) + (I)*(B(6,6))} } |
| | |
| | </code> |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of spherical Harmonics** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ |
| | ^$ {Y_{-2}^{(2)}} $|$ -\sqrt{\frac{3}{7}} (A(1,1)+i B(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)+i B(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)+i B(5,1)) $|$ 0 $|$ \frac{33 \sqrt{35} (A(1,1)-i B(1,1))-22 \sqrt{210} (A(3,1)-i B(3,1))+25 \sqrt{21} (A(5,1)-i B(5,1))}{1155} $|$ 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 $|$ \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i B(5,5)) $| |
| | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ -\sqrt{\frac{2}{7}} (A(1,1)+i B(1,1))-\frac{A(3,1)+i B(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (A(5,1)+i B(5,1)) $|$ 0 $|$ \frac{33 \sqrt{105} (A(1,1)-i B(1,1))-11 \sqrt{70} (A(3,1)-i B(3,1))-100 \sqrt{7} (A(5,1)-i B(5,1))}{1155} $|$ 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 $|$ -\sqrt{\frac{6}{35}} (A(1,1)+i B(1,1))-\frac{A(3,1)+i B(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (A(5,1)+i B(5,1)) $|$ 0 $|$ \frac{1}{385} \left(11 \sqrt{210} (A(1,1)-i B(1,1))+11 \sqrt{35} (A(3,1)-i B(3,1))+25 \sqrt{14} (A(5,1)-i B(5,1))\right) $|$ 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 $|$ -\sqrt{\frac{3}{35}} (A(1,1)+i B(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (A(3,1)+i B(3,1))+\frac{20 (A(5,1)+i B(5,1))}{33 \sqrt{7}} $|$ 0 $|$ \frac{1}{231} \left(33 \sqrt{14} (A(1,1)-i B(1,1))+11 \sqrt{21} (A(3,1)-i B(3,1))-2 \sqrt{210} (A(5,1)-i B(5,1))\right) $|$ 0 $| |
| | ^$ {Y_{2}^{(2)}} $|$ -\frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)+i B(5,5)) $|$ 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 $|$ -\frac{A(1,1)+i B(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (A(3,1)+i B(3,1))-\frac{5 (A(5,1)+i B(5,1))}{11 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{7}} (A(1,1)-i B(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)-i B(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)-i B(5,1)) $| |
| | |
| | |
| | ### |
| | |
| | </hidden> |
| | <hidden **The Hamiltonian on a basis of symmetric functions** > |
| | |
| | ### |
| | |
| | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ |
| | ^$ d_{x^2-y^2} $|$ 0 $|$ \frac{-99 \sqrt{210} A(1,1)+121 \sqrt{35} A(3,1)-5 \left(11 \sqrt{21} A(3,3)+10 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)+35 \sqrt{15} A(5,5)\right)}{2310} $|$ \frac{-99 \sqrt{210} B(1,1)+121 \sqrt{35} B(3,1)+55 \sqrt{21} B(3,3)-50 \sqrt{14} B(5,1)-175 \sqrt{3} B(5,3)-175 \sqrt{15} B(5,5)}{2310} $|$ 0 $|$ \frac{1}{462} \left(33 \sqrt{14} A(1,1)+11 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)-2 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)+105 A(5,5)\right) $|$ \frac{1}{462} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)-11 \sqrt{35} B(3,3)+2 \sqrt{210} B(5,1)+35 \sqrt{5} B(5,3)-105 B(5,5)\right) $|$ 0 $| |
| | ^$ d_{3z^2-r^2} $|$ 0 $|$ \frac{99 \sqrt{70} A(1,1)+33 \sqrt{105} A(3,1)+275 \sqrt{7} A(3,3)+75 \sqrt{42} A(5,1)-350 A(5,3)}{2310} $|$ \frac{-99 \sqrt{70} B(1,1)-33 \sqrt{105} B(3,1)+275 \sqrt{7} B(3,3)-75 \sqrt{42} B(5,1)-350 B(5,3)}{2310} $|$ 0 $|$ \frac{1}{462} \left(33 \sqrt{42} A(1,1)+33 \sqrt{7} A(3,1)-11 \sqrt{105} A(3,3)+15 \sqrt{70} A(5,1)+14 \sqrt{15} A(5,3)\right) $|$ \frac{1}{462} \left(33 \sqrt{42} B(1,1)+33 \sqrt{7} B(3,1)+11 \sqrt{105} B(3,3)+15 \sqrt{70} B(5,1)-14 \sqrt{15} B(5,3)\right) $|$ 0 $| |
| | ^$ d_{\text{yz}} $|$ \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)+28 A(5,3)\right)\right) $|$ 0 $|$ 0 $|$ -\sqrt{\frac{6}{35}} B(1,1)+\frac{2 B(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} B(5,1) $|$ 0 $|$ 0 $|$ \frac{1}{231} \left(-33 \sqrt{14} B(1,1)-11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)+2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $| |
| | ^$ d_{\text{xz}} $|$ \frac{1}{231} \left(33 \sqrt{14} B(1,1)+11 \sqrt{21} B(3,1)+\sqrt{5} \left(11 \sqrt{7} B(3,3)-2 \sqrt{42} B(5,1)+28 B(5,3)\right)\right) $|$ 0 $|$ 0 $|$ \sqrt{\frac{6}{35}} A(1,1)-\frac{2 A(3,1)}{3 \sqrt{35}}-\frac{20}{33} \sqrt{\frac{2}{7}} A(5,1) $|$ 0 $|$ 0 $|$ \frac{1}{231} \left(-33 \sqrt{14} A(1,1)-11 \sqrt{21} A(3,1)+\sqrt{5} \left(-11 \sqrt{7} A(3,3)+2 \sqrt{42} A(5,1)-28 A(5,3)\right)\right) $| |
| | ^$ d_{\text{xy}} $|$ 0 $|$ \frac{-66 \sqrt{210} B(1,1)-11 \sqrt{35} B(3,1)+5 \left(11 \sqrt{21} B(3,3)+5 \sqrt{14} B(5,1)-35 \sqrt{3} B(5,3)+35 \sqrt{15} B(5,5)\right)}{2310} $|$ \frac{66 \sqrt{210} A(1,1)+11 \sqrt{35} A(3,1)+5 \left(11 \sqrt{21} A(3,3)-5 \sqrt{14} A(5,1)-35 \sqrt{3} A(5,3)-35 \sqrt{15} A(5,5)\right)}{2310} $|$ 0 $|$ \frac{1}{462} \left(66 \sqrt{14} B(1,1)-33 \sqrt{21} B(3,1)+11 \sqrt{35} B(3,3)+3 \sqrt{210} B(5,1)-35 \sqrt{5} B(5,3)-105 B(5,5)\right) $|$ \frac{1}{462} \left(66 \sqrt{14} A(1,1)-33 \sqrt{21} A(3,1)-11 \sqrt{35} A(3,3)+3 \sqrt{210} A(5,1)+35 \sqrt{5} A(5,3)-105 A(5,5)\right) $|$ 0 $| |
| | |
| | |
| | ### |
| |
| | </hidden> |
| |
| ===== Table of several point groups ===== | ===== Table of several point groups ===== |
