Differences

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--- physics_chemistry:point_groups:cs:orientation_z [2018/03/29 20:47] – Maurits W. Haverkort
+++ physics_chemistry:point_groups:cs:orientation_z [2018/03/30 10:01] – Maurits W. Haverkort
@@ Line 91: / Line 91: @@
 ###
-==== Input format suitable for Mathematica (Quanty.nb) ====
+==== Expansion ====
 ###
@@ Line 97: / Line 97: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
+###
+==== 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>
 ###
@@ Line 135: / Line 147: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 299: / Line 311: @@
 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 <hiddenSwitch expand all>
 ###
@@ Line 304: / Line 324: @@
 ==== Potential for s orbitals ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 312: / Line 332: @@
 & \text{True}
 \end{cases}$$
+###
+</hidden>
+<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+###
+<code Quanty Akm_Cs_Z.Quanty.nb>
+Akm[k_,m_]:=Piecewise[{{Eag, k == 0 && m == 0}}, 0]
+</code>
 ###
@@ Line 327: / Line 360: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
 ###
@@ Line 337: / Line 371: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of symmetric functions** >
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
 ###
@@ Line 347: / Line 382: @@
 ###
-</hidden><hidden **Rotation matrix used** >
+</hidden>
+<hidden **Rotation matrix used** >
 ###
- TODO
+|  $  $  ^  $ {Y_{0}^{(0)}} $  ^
+^$ \text{s} $|$ 1 $|
 ###
-</hidden><hidden **Irriducible representations and their onsite energy** >
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
 ###
- TODO
+^ ^$$\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 }}$$ | ::: |
 ###
-</hidden>==== Potential for p orbitals ====
+</hidden>
+==== Potential for p orbitals ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 377: / Line 420: @@
  \frac{5 (\text{Epxpx}-2 i \text{Epypx}-\text{Epypy})}{2 \sqrt{6}} & k=2\land m=2
 \end{cases}$$
+###
+</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>
 ###
@@ Line 397: / Line 453: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
 ###
@@ Line 409: / Line 466: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of symmetric functions** >
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
 ###
@@ Line 421: / Line 479: @@
 ###
-</hidden><hidden **Rotation matrix used** >
+</hidden>
+<hidden **Rotation matrix used** >
 ###
- TODO
+|  $  $  ^  $ {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** >
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
 ###
- TODO
+^ ^$$\text{Epxpx}$$ | {{: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 for d orbitals ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 460: / Line 534: @@
  \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>
 ###
@@ Line 489: / Line 576: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</hidden>
+<hidden **The Hamiltonian on a basis of spherical Harmonics** >
 ###
@@ Line 503: / Line 591: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of symmetric functions** >
+</hidden>
+<hidden **The Hamiltonian on a basis of symmetric functions** >
 ###
@@ Line 517: / Line 606: @@
 ###
-</hidden><hidden **Rotation matrix used** >
+</hidden>
+<hidden **Rotation matrix used** >
 ###
- TODO
+|  $  $  ^  $ {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** >
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
 ###
- TODO
+^ ^$$\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$$ | ::: |
 ###
-</hidden>==== Potential for f orbitals ====
+</hidden>
+==== Potential for f orbitals ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 541: / Line 654: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
+###
+</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>
 ###
@@ Line 579: / Line 705: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 611: / Line 737: @@
 ###
-</hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i \text{Ap}(6,2)) $|$ 0 $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)-i \text{Ap}(6,4))\right) $|$ 0 $|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (A(6,6)-i \text{Ap}(6,6)) $|
+^$ {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 \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i \text{Ap}(6,2)) $|$ 0 $|$ \frac{1}{429} \sqrt{14} \left(13 \sqrt{5} (A(4,4)-i \text{Ap}(4,4))+30 (A(6,4)-i \text{Ap}(6,4))\right) $|$ 0 $|
+^$ {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 \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i \text{Ap}(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 \text{Ap}(2,2))-65 \sqrt{10} (A(4,2)-i \text{Ap}(4,2))-25 \sqrt{105} (A(6,2)-i \text{Ap}(6,2))\right)}{2145} $|$ 0 $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)-i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)-i \text{Ap}(6,4))\right) $|
+^$ {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 \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i \text{Ap}(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 \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)-i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)-i \text{Ap}(6,2)) $|$ 0 $|
+^$ {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 \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)+i \text{Ap}(6,4))\right) $|$ 0 $|$ \frac{2 \left(-143 \sqrt{6} (A(2,2)+i \text{Ap}(2,2))-65 \sqrt{10} (A(4,2)+i \text{Ap}(4,2))-25 \sqrt{105} (A(6,2)+i \text{Ap}(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 \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)-i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)-i \text{Ap}(6,2)) $|
+^$ {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 \text{Ap}(4,4))+30 (A(6,4)+i \text{Ap}(6,4))\right) $|$ 0 $|$ -\frac{2 (A(2,2)+i \text{Ap}(2,2))}{3 \sqrt{5}}-\frac{A(4,2)+i \text{Ap}(4,2)}{11 \sqrt{3}}+\frac{20}{429} \sqrt{14} (A(6,2)+i \text{Ap}(6,2)) $|$ 0 $|$ A(0,0)-\frac{7}{33} A(4,0)+\frac{10}{143} A(6,0) $|$ 0 $|
+^$ {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 \text{Ap}(6,6)) $|$ 0 $|$ \frac{1}{143} \sqrt{\frac{14}{3}} \left(13 (A(4,4)+i \text{Ap}(4,4))-5 \sqrt{5} (A(6,4)+i \text{Ap}(6,4))\right) $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{5}} (A(2,2)+i \text{Ap}(2,2))+\frac{1}{11} \sqrt{6} (A(4,2)+i \text{Ap}(4,2))-\frac{10}{429} \sqrt{7} (A(6,2)+i \text{Ap}(6,2)) $|$ 0 $|$ A(0,0)-\frac{1}{3} A(2,0)+\frac{1}{11} A(4,0)-\frac{5}{429} A(6,0) $|
+^$ {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** >
+</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} \text{Ap}(2,2)+65 \sqrt{6} \text{Ap}(4,2)-200 \sqrt{7} \text{Ap}(6,2)}{2145} $|$ 0 $|$ 0 $|$ -\frac{1}{429} \sqrt{14} \left(13 \sqrt{5} \text{Ap}(4,4)+30 \text{Ap}(6,4)\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} \text{Ap}(2,2)-780 \sqrt{10} \text{Ap}(4,2)-25 \sqrt{105} \text{Ap}(6,2)+125 \sqrt{231} \text{Ap}(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} \text{Ap}(2,2)+5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)-20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(6,6)\right)\right)}{8580} $|$ 0 $|
+^$ 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} \text{Ap}(2,2)-780 \sqrt{10} \text{Ap}(4,2)-25 \sqrt{105} \text{Ap}(6,2)+125 \sqrt{231} \text{Ap}(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} \text{Ap}(2,2)-5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(-52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)+20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(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_{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} \text{Ap}(2,2)+65 \sqrt{6} \text{Ap}(4,2)-200 \sqrt{7} \text{Ap}(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(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} \text{Ap}(2,2)-5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(-52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)+20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(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} \text{Ap}(2,2)-52 \sqrt{10} \text{Ap}(4,2)+35 \sqrt{105} \text{Ap}(6,2)-15 \sqrt{231} \text{Ap}(6,6)}{1716} $|$ 0 $|
+^$ 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} \text{Ap}(2,2)+5 \left(104 \sqrt{6} \text{Ap}(4,2)+\sqrt{7} \left(52 \sqrt{6} \text{Ap}(4,4)+65 \text{Ap}(6,2)-20 \sqrt{30} \text{Ap}(6,4)+15 \sqrt{55} \text{Ap}(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} \text{Ap}(2,2)-52 \sqrt{10} \text{Ap}(4,2)+35 \sqrt{105} \text{Ap}(6,2)-15 \sqrt{231} \text{Ap}(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_{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} \text{Ap}(4,4)+30 \text{Ap}(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) $|
+^$ 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** >
+</hidden>
+<hidden **Rotation matrix used** >
 ###
- TODO
+|  $  $  ^  $ {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** >
+</hidden>
+<hidden **Irriducible representations and their onsite energy** >
 ###
- TODO
+^ ^$$\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)$$ | ::: |
 ###
-</hidden>===== Coupling between two shells =====
+</hidden>
+===== Coupling between two shells =====
+###
+Click on one of the subsections to expand it or <hiddenSwitch expand all>
+###
 ==== Potential for s-p orbital mixing ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 669: / Line 837: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
 ###
-<\hidden><hidden **Input format suitable for Quanty** >
+</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** >
 ###
@@ Line 707: / Line 888: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 739: / Line 920: @@
 ###
-<\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(1,1)}{\sqrt{3}} $|$ 0 $|$ \frac{A(1,1)-i \text{Ap}(1,1)}{\sqrt{3}} $|
+^$ {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** >
+</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}} \text{Ap}(1,1) $|$ 0 $|
+^$ \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 for s-d orbital mixing ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 767: / Line 951: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
 ###
-<\hidden><hidden **Input format suitable for Quanty** >
+</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** >
 ###
@@ Line 805: / Line 1002: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 837: / Line 1034: @@
 ###
-<\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(2,2)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,0)}{\sqrt{5}} $|$ 0 $|$ \frac{A(2,2)-i \text{Ap}(2,2)}{\sqrt{5}} $|
+^$ {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** >
+</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}} \text{Ap}(2,2) $|
+^$ \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 for s-f orbital mixing ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 865: / Line 1065: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
 ###
-<\hidden><hidden **Input format suitable for Quanty** >
+</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** >
 ###
@@ Line 903: / Line 1116: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 935: / Line 1148: @@
 ###
-<\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(3,3)}{\sqrt{7}} $|$ 0 $|$ -\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,1)-i \text{Ap}(3,1)}{\sqrt{7}} $|$ 0 $|$ \frac{A(3,3)-i \text{Ap}(3,3)}{\sqrt{7}} $|
+^$ {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** >
+</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} \text{Ap}(3,1)+\sqrt{5} \text{Ap}(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} \text{Ap}(3,1)-\sqrt{21} \text{Ap}(3,3)\right) $|$ 0 $|
+^$ \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 for p-d orbital mixing ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 963: / Line 1179: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
 ###
-<\hidden><hidden **Input format suitable for Quanty** >
+</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** >
 ###
@@ Line 1001: / Line 1230: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 1033: / Line 1262: @@
 ###
-<\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(3,1))-\sqrt{\frac{2}{5}} (A(1,1)+i \text{Ap}(1,1)) $|$ 0 $|$ \frac{1}{105} \left(7 \sqrt{15} (A(1,1)-i \text{Ap}(1,1))-9 \sqrt{10} (A(3,1)-i \text{Ap}(3,1))\right) $|$ 0 $|$ -\frac{3}{7} (A(3,3)-i \text{Ap}(3,3)) $|
+^$ {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 \text{Ap}(1,1)}{\sqrt{5}}-\frac{2}{7} \sqrt{\frac{6}{5}} (A(3,1)+i \text{Ap}(3,1)) $|$ 0 $|$ \frac{7 A(1,1)+2 \sqrt{6} A(3,1)-i \left(7 \text{Ap}(1,1)+2 \sqrt{6} \text{Ap}(3,1)\right)}{7 \sqrt{5}} $|$ 0 $|
+^$ {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 \text{Ap}(3,3)) $|$ 0 $|$ \frac{3}{7} \sqrt{\frac{2}{5}} (A(3,1)+i \text{Ap}(3,1))-\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{15}} $|$ 0 $|$ \sqrt{\frac{2}{5}} (A(1,1)-i \text{Ap}(1,1))-\frac{1}{7} \sqrt{\frac{3}{5}} (A(3,1)-i \text{Ap}(3,1)) $|
+^$ {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** >
+</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}} \text{Ap}(1,1)-\frac{1}{7} \sqrt{\frac{3}{5}} \text{Ap}(3,1)+\frac{3}{7} \text{Ap}(3,3) $|
+^$ 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} \text{Ap}(1,1)+\sqrt{15} \text{Ap}(3,1)+15 \text{Ap}(3,3)\right) $|$ \frac{6 \text{Ap}(3,1)}{7 \sqrt{5}}-\sqrt{\frac{2}{15}} \text{Ap}(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_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}} \text{Ap}(1,1)+\frac{4}{7} \sqrt{\frac{3}{5}} \text{Ap}(3,1) $|$ -\frac{1}{7} \sqrt{\frac{2}{5}} \left(7 A(1,1)+2 \sqrt{6} A(3,1)\right) $|$ 0 $|
+^$ 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 for p-f orbital mixing ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 1065: / Line 1297: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
 ###
-<\hidden><hidden **Input format suitable for Quanty** >
+</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** >
 ###
@@ Line 1103: / Line 1348: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 1135: / Line 1380: @@
 ###
-<\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(2,2))}{\sqrt{35}}-\frac{A(4,2)+i \text{Ap}(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 \text{Ap}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)-i \text{Ap}(4,2)) $|$ 0 $|$ -\frac{2 (A(4,4)-i \text{Ap}(4,4))}{3 \sqrt{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 \text{Ap}(2,2))+\frac{2 (A(4,2)+i \text{Ap}(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 \text{Ap}(2,2))+\frac{2 (A(4,2)-i \text{Ap}(4,2))}{3 \sqrt{7}} $|$ 0 $|
+^$ {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 \text{Ap}(4,4))}{3 \sqrt{3}} $|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} (A(2,2)+i \text{Ap}(2,2))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(4,2)+i \text{Ap}(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 \text{Ap}(2,2))}{\sqrt{35}}-\frac{A(4,2)-i \text{Ap}(4,2)}{3 \sqrt{21}} $|
+^$ {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** >
+</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} \text{Ap}(2,2)+5 \sqrt{30} \left(\sqrt{7} \text{Ap}(4,2)+7 \text{Ap}(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}} \text{Ap}(2,2)-\frac{\text{Ap}(4,2)}{\sqrt{14}}+\frac{\text{Ap}(4,4)}{3 \sqrt{2}} $|$ 0 $|
+^$ 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} \text{Ap}(2,2)+5 \sqrt{30} \left(\sqrt{7} \text{Ap}(4,2)-7 \text{Ap}(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}} \text{Ap}(2,2)+\frac{\text{Ap}(4,2)}{\sqrt{14}}+\frac{\text{Ap}(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_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}} \text{Ap}(2,2)-\frac{2}{3} \sqrt{\frac{2}{7}} \text{Ap}(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) $|
+^$ 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 for d-f orbital mixing ====
-<hidden **Input format suitable for Mathematica (Quanty.nb)** >
+<hidden **Potential parameterized with onsite energies of irriducible representations** >
 ###
@@ Line 1167: / Line 1415: @@
  $$A_{k,m} = \begin{cases}
  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(2,2)+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(4,2)+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(6,2)+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}$$
 ###
-<\hidden><hidden **Input format suitable for Quanty** >
+</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** >
 ###
@@ Line 1205: / Line 1466: @@
 Akm = {{0, 0, A(0,0)} ,
-       {1,-1, (-1)*(A(1,1)) + (I)*(Ap(1,1))} ,
+       {1,-1, (-1)*(A(1,1)) + (I)*(B(1,1))} ,
-       {1, 1, A(1,1) + (I)*(Ap(1,1))} ,
+       {1, 1, A(1,1) + (I)*(B(1,1))} ,
        {2, 0, A(2,0)} ,
-       {2,-2, A(2,2) + (-I)*(Ap(2,2))} ,
+       {2,-2, A(2,2) + (-I)*(B(2,2))} ,
-       {2, 2, A(2,2) + (I)*(Ap(2,2))} ,
+       {2, 2, A(2,2) + (I)*(B(2,2))} ,
-       {3,-1, (-1)*(A(3,1)) + (I)*(Ap(3,1))} ,
+       {3,-1, (-1)*(A(3,1)) + (I)*(B(3,1))} ,
-       {3, 1, A(3,1) + (I)*(Ap(3,1))} ,
+       {3, 1, A(3,1) + (I)*(B(3,1))} ,
-       {3,-3, (-1)*(A(3,3)) + (I)*(Ap(3,3))} ,
+       {3,-3, (-1)*(A(3,3)) + (I)*(B(3,3))} ,
-       {3, 3, A(3,3) + (I)*(Ap(3,3))} ,
+       {3, 3, A(3,3) + (I)*(B(3,3))} ,
        {4, 0, A(4,0)} ,
-       {4,-2, A(4,2) + (-I)*(Ap(4,2))} ,
+       {4,-2, A(4,2) + (-I)*(B(4,2))} ,
-       {4, 2, A(4,2) + (I)*(Ap(4,2))} ,
+       {4, 2, A(4,2) + (I)*(B(4,2))} ,
-       {4,-4, A(4,4) + (-I)*(Ap(4,4))} ,
+       {4,-4, A(4,4) + (-I)*(B(4,4))} ,
-       {4, 4, A(4,4) + (I)*(Ap(4,4))} ,
+       {4, 4, A(4,4) + (I)*(B(4,4))} ,
-       {5,-1, (-1)*(A(5,1)) + (I)*(Ap(5,1))} ,
+       {5,-1, (-1)*(A(5,1)) + (I)*(B(5,1))} ,
-       {5, 1, A(5,1) + (I)*(Ap(5,1))} ,
+       {5, 1, A(5,1) + (I)*(B(5,1))} ,
-       {5,-3, (-1)*(A(5,3)) + (I)*(Ap(5,3))} ,
+       {5,-3, (-1)*(A(5,3)) + (I)*(B(5,3))} ,
-       {5, 3, A(5,3) + (I)*(Ap(5,3))} ,
+       {5, 3, A(5,3) + (I)*(B(5,3))} ,
-       {5,-5, (-1)*(A(5,5)) + (I)*(Ap(5,5))} ,
+       {5,-5, (-1)*(A(5,5)) + (I)*(B(5,5))} ,
-       {5, 5, A(5,5) + (I)*(Ap(5,5))} ,
+       {5, 5, A(5,5) + (I)*(B(5,5))} ,
        {6, 0, A(6,0)} ,
-       {6,-2, A(6,2) + (-I)*(Ap(6,2))} ,
+       {6,-2, A(6,2) + (-I)*(B(6,2))} ,
-       {6, 2, A(6,2) + (I)*(Ap(6,2))} ,
+       {6, 2, A(6,2) + (I)*(B(6,2))} ,
-       {6,-4, A(6,4) + (-I)*(Ap(6,4))} ,
+       {6,-4, A(6,4) + (-I)*(B(6,4))} ,
-       {6, 4, A(6,4) + (I)*(Ap(6,4))} ,
+       {6, 4, A(6,4) + (I)*(B(6,4))} ,
-       {6,-6, A(6,6) + (-I)*(Ap(6,6))} ,
+       {6,-6, A(6,6) + (-I)*(B(6,6))} ,
-       {6, 6, A(6,6) + (I)*(Ap(6,6))} }
+       {6, 6, A(6,6) + (I)*(B(6,6))} }
 </code>
@@ Line 1237: / Line 1498: @@
 ###
-<\hidden><hidden **The Hamiltonian on a basis of spherical Harmonics** >
+</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 \text{Ap}(1,1))+\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)+i \text{Ap}(3,1))-\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)+i \text{Ap}(5,1)) $|$ 0 $|$ \frac{33 \sqrt{35} (A(1,1)-i \text{Ap}(1,1))-22 \sqrt{210} (A(3,1)-i \text{Ap}(3,1))+25 \sqrt{21} (A(5,1)-i \text{Ap}(5,1))}{1155} $|$ 0 $|$ \frac{5}{33} \sqrt{2} (A(5,3)-i \text{Ap}(5,3))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)-i \text{Ap}(3,3)) $|$ 0 $|$ \frac{5}{11} \sqrt{\frac{2}{3}} (A(5,5)-i \text{Ap}(5,5)) $|
+^$ {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 \text{Ap}(1,1))-\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{21}}+\frac{2}{11} \sqrt{\frac{10}{21}} (A(5,1)+i \text{Ap}(5,1)) $|$ 0 $|$ \frac{33 \sqrt{105} (A(1,1)-i \text{Ap}(1,1))-11 \sqrt{70} (A(3,1)-i \text{Ap}(3,1))-100 \sqrt{7} (A(5,1)-i \text{Ap}(5,1))}{1155} $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i \text{Ap}(3,3))-\frac{4}{33} \sqrt{5} (A(5,3)-i \text{Ap}(5,3)) $|$ 0 $|
+^$ {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 \text{Ap}(3,3))-\frac{2}{33} \sqrt{5} (A(5,3)+i \text{Ap}(5,3)) $|$ 0 $|$ -\sqrt{\frac{6}{35}} (A(1,1)+i \text{Ap}(1,1))-\frac{A(3,1)+i \text{Ap}(3,1)}{\sqrt{35}}-\frac{5}{11} \sqrt{\frac{2}{7}} (A(5,1)+i \text{Ap}(5,1)) $|$ 0 $|$ \frac{1}{385} \left(11 \sqrt{210} (A(1,1)-i \text{Ap}(1,1))+11 \sqrt{35} (A(3,1)-i \text{Ap}(3,1))+25 \sqrt{14} (A(5,1)-i \text{Ap}(5,1))\right) $|$ 0 $|$ \frac{2}{33} \sqrt{5} (A(5,3)-i \text{Ap}(5,3))-\frac{1}{3} \sqrt{\frac{5}{7}} (A(3,3)-i \text{Ap}(3,3)) $|
+^$ {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 \text{Ap}(3,3))+\frac{4}{33} \sqrt{5} (A(5,3)+i \text{Ap}(5,3)) $|$ 0 $|$ -\sqrt{\frac{3}{35}} (A(1,1)+i \text{Ap}(1,1))+\frac{1}{3} \sqrt{\frac{2}{35}} (A(3,1)+i \text{Ap}(3,1))+\frac{20 (A(5,1)+i \text{Ap}(5,1))}{33 \sqrt{7}} $|$ 0 $|$ \frac{1}{231} \left(33 \sqrt{14} (A(1,1)-i \text{Ap}(1,1))+11 \sqrt{21} (A(3,1)-i \text{Ap}(3,1))-2 \sqrt{210} (A(5,1)-i \text{Ap}(5,1))\right) $|$ 0 $|
+^$ {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 \text{Ap}(5,5)) $|$ 0 $|$ \frac{1}{3} \sqrt{\frac{2}{7}} (A(3,3)+i \text{Ap}(3,3))-\frac{5}{33} \sqrt{2} (A(5,3)+i \text{Ap}(5,3)) $|$ 0 $|$ -\frac{A(1,1)+i \text{Ap}(1,1)}{\sqrt{35}}+2 \sqrt{\frac{2}{105}} (A(3,1)+i \text{Ap}(3,1))-\frac{5 (A(5,1)+i \text{Ap}(5,1))}{11 \sqrt{21}} $|$ 0 $|$ \sqrt{\frac{3}{7}} (A(1,1)-i \text{Ap}(1,1))-\frac{1}{3} \sqrt{\frac{2}{7}} (A(3,1)-i \text{Ap}(3,1))+\frac{1}{33} \sqrt{\frac{5}{7}} (A(5,1)-i \text{Ap}(5,1)) $|
+^$ {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** >
+</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} \text{Ap}(1,1)+121 \sqrt{35} \text{Ap}(3,1)+55 \sqrt{21} \text{Ap}(3,3)-50 \sqrt{14} \text{Ap}(5,1)-175 \sqrt{3} \text{Ap}(5,3)-175 \sqrt{15} \text{Ap}(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} \text{Ap}(1,1)-11 \sqrt{21} \text{Ap}(3,1)-11 \sqrt{35} \text{Ap}(3,3)+2 \sqrt{210} \text{Ap}(5,1)+35 \sqrt{5} \text{Ap}(5,3)-105 \text{Ap}(5,5)\right) $|$ 0 $|
+^$ 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} \text{Ap}(1,1)-33 \sqrt{105} \text{Ap}(3,1)+275 \sqrt{7} \text{Ap}(3,3)-75 \sqrt{42} \text{Ap}(5,1)-350 \text{Ap}(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} \text{Ap}(1,1)+33 \sqrt{7} \text{Ap}(3,1)+11 \sqrt{105} \text{Ap}(3,3)+15 \sqrt{70} \text{Ap}(5,1)-14 \sqrt{15} \text{Ap}(5,3)\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}} \text{Ap}(1,1)+\frac{2 \text{Ap}(3,1)}{3 \sqrt{35}}+\frac{20}{33} \sqrt{\frac{2}{7}} \text{Ap}(5,1) $|$ 0 $|$ 0 $|$ \frac{1}{231} \left(-33 \sqrt{14} \text{Ap}(1,1)-11 \sqrt{21} \text{Ap}(3,1)+\sqrt{5} \left(11 \sqrt{7} \text{Ap}(3,3)+2 \sqrt{42} \text{Ap}(5,1)+28 \text{Ap}(5,3)\right)\right) $|
+^$ 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} \text{Ap}(1,1)+11 \sqrt{21} \text{Ap}(3,1)+\sqrt{5} \left(11 \sqrt{7} \text{Ap}(3,3)-2 \sqrt{42} \text{Ap}(5,1)+28 \text{Ap}(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{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} \text{Ap}(1,1)-11 \sqrt{35} \text{Ap}(3,1)+5 \left(11 \sqrt{21} \text{Ap}(3,3)+5 \sqrt{14} \text{Ap}(5,1)-35 \sqrt{3} \text{Ap}(5,3)+35 \sqrt{15} \text{Ap}(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} \text{Ap}(1,1)-33 \sqrt{21} \text{Ap}(3,1)+11 \sqrt{35} \text{Ap}(3,3)+3 \sqrt{210} \text{Ap}(5,1)-35 \sqrt{5} \text{Ap}(5,3)-105 \text{Ap}(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 $|
+^$ 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 $|
@@ Line 1266: / Line 1529: @@
 </hidden>
 ===== Table of several point groups =====
@@ Line 1288: / Line 1552: @@
 ###

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