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--- physics_chemistry:point_groups:d3d:orientation_zx [2018/04/05 00:05] – Maurits W. Haverkort
+++ physics_chemistry:point_groups:d3d:orientation_zx [2018/09/06 13:01] (current) – Maurits W. Haverkort
@@ Line 1: / Line 1: @@
+~~CLOSETOC~~
 ====== Orientation Zx ======
 ###
-The point group D3d is a subgroup of Oh. Many materials of relavance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the sttes in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irriducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irriducible representation and two that belong to the eg irricucible representation. We label the eg orbitals that decend from the eg irriducible representation in Oh symmetry eg\[Sigma] and the eg orbitals that decend from the t2g irriducible representation eg\[Pi] orbitals. (The mixing is given by the parameter Meg.)
+The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry eg$\sigma$ and the eg orbitals that descend from the t2g irreducible representation eg$\pi$ orbitals. (The mixing is given by the parameter Meg.)
-As one can see in the list of supergroups of D3d, there are two different orientations of Oh that are a supergroup of this orientation of D3d. The different orientations of Oh with respect to D3d do however change the definitions of the eg\[Pi] and eg\[Sigma] orbitals. We inlcude three different representations of the orbitals and potentials for each setting of D3d symmetry. The orientation without aditional letter takes the tesseral harmonics as a basis. This basis does not relate to the states in Oh symmetry. The orientation with an aditional A or B relate to the two differnt supergroup representations of the Oh point group.
+As one can see in the list of supergroups of D3d, there are two different orientations of Oh that are a supergroup of this orientation of D3d. The different orientations of Oh with respect to D3d do however change the definitions of the eg$\pi$ and eg$\sigma$ orbitals. We include three different representations of the orbitals and potentials for each setting of D3d symmetry. The orientation without additional letter takes the tesseral harmonics as a basis. This basis does not relate to the states in Oh symmetry. The orientation with an additional A or B relate to the two different supergroup representations of the Oh point group.
 ###
@@ Line 387: / Line 389: @@
 ###
-^ ^ $\text{Ea1g}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_0_1.png?50}} |
+^ ^ $\text{Ea1g}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_0_1.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2 \sqrt{\pi }}$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2 \sqrt{\pi }}$  | ::: |
@@ Line 478: / Line 480: @@
 ###
-^ ^ $\text{Eeu}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_1_1.png?50}} |
+^ ^ $\text{Eeu}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_1_1.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \sin (\phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} y$  | ::: |
-^ ^ $\text{Ea2u}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_1_2.png?50}} |
+^ ^ $\text{Ea2u}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_1_2.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} z$  | ::: |
-^ ^ $\text{Eeu}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_1_3.png?50}} |
+^ ^ $\text{Eeu}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_1_3.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} \sin (\theta ) \cos (\phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{3}{\pi }} x$  | ::: |
@@ Line 586: / Line 588: @@
 ###
-^ ^ $\text{Eeg2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_1.png?50}} |
+^ ^ $\text{Eeg2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_1.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \sin (2 \phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} x y$  | ::: |
-^ ^ $\text{Eeg1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_2.png?50}} |
+^ ^ $\text{Eeg1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_2.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \sin (\phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} y z$  | ::: |
-^ ^ $\text{Ea1g}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_3.png?50}} |
+^ ^ $\text{Ea1g}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_3.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{8} \sqrt{\frac{5}{\pi }} (3 \cos (2 \theta )+1)$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 z^2-1\right)$  | ::: |
-^ ^ $\text{Eeg1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_4.png?50}} |
+^ ^ $\text{Eeg1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_4.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin (2 \theta ) \cos (\phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{15}{\pi }} x z$  | ::: |
-^ ^ $\text{Eeg2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_5.png?50}} |
+^ ^ $\text{Eeg2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_2_5.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \sin ^2(\theta ) \cos (2 \phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{15}{\pi }} \left(x^2-y^2\right)$  | ::: |
@@ Line 714: / Line 716: @@
 ###
-^ ^ $\text{Ea2uA}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_1.png?50}} |
+^ ^ $\text{Ea2uA}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_1.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \sin (3 \phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $-\frac{1}{4} \sqrt{\frac{35}{2 \pi }} y \left(y^2-3 x^2\right)$  | ::: |
-^ ^ $\text{Eeu2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_2.png?50}} |
+^ ^ $\text{Eeu2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_2.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \sin (2 \phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{2} \sqrt{\frac{105}{\pi }} x y z$  | ::: |
-^ ^ $\text{Eeu1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_3.png?50}} |
+^ ^ $\text{Eeu1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_3.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{8} \sqrt{\frac{21}{2 \pi }} \sin (\theta ) (5 \cos (2 \theta )+3) \sin (\phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{21}{2 \pi }} y \left(5 z^2-1\right)$  | ::: |
-^ ^ $\text{Ea2uB}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_4.png?50}} |
+^ ^ $\text{Ea2uB}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_4.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{7}{\pi }} (3 \cos (\theta )+5 \cos (3 \theta ))$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(5 z^2-3\right)$  | ::: |
-^ ^ $\text{Eeu1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_5.png?50}} |
+^ ^ $\text{Eeu1}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_5.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{16} \sqrt{\frac{21}{2 \pi }} (\sin (\theta )+5 \sin (3 \theta )) \cos (\phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{21}{2 \pi }} x \left(5 z^2-1\right)$  | ::: |
-^ ^ $\text{Eeu2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_6.png?50}} |
+^ ^ $\text{Eeu2}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_6.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} \sin ^2(\theta ) \cos (\theta ) \cos (2 \phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{105}{\pi }} z \left(x^2-y^2\right)$  | ::: |
-^ ^ $\text{Ea1u}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_7.png?50}} |
+^ ^ $\text{Ea1u}$  | {{:physics_chemistry:pointgroup:d3d_zx_orb_3_7.png?150}} |
 | $\psi(\theta,\phi)=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{35}{2 \pi }} \sin ^3(\theta ) \cos (3 \phi )$  | ::: |
 | $\psi(\hat{x},\hat{y},\hat{z})=\phantom{\sqrt{\frac{1}{1}}}$  | $\frac{1}{4} \sqrt{\frac{35}{2 \pi }} x \left(x^2-3 y^2\right)$  | ::: |

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