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| physics_chemistry:point_groups:d3d:orientation_zy_a [2018/03/21 18:40] – created Stefano Agrestini | physics_chemistry:point_groups:d3d:orientation_zy_a [2018/09/06 13:56] (current) – Maurits W. Haverkort | ||
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| + | ~~CLOSETOC~~ | ||
| + | |||
| ====== Orientation Zy_A ====== | ====== Orientation Zy_A ====== | ||
| ### | ### | ||
| - | alligned paragraph text | + | 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.) |
| ### | ### | ||
| - | ===== Example ===== | + | ### |
| + | 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. | ||
| + | ### | ||
| ### | ### | ||
| - | description text | + | The parameterization A of the orientation Zy is related to the orientation Sqrt[2]01z of the Oh pointgroup. |
| ### | ### | ||
| - | ==== Input ==== | + | ===== Symmetry Operations ===== |
| - | <code Quanty | + | |
| - | -- some example code | + | ### |
| + | |||
| + | In the D3d Point Group, with orientation Zy_A there are the following symmetry operations | ||
| + | |||
| + | ### | ||
| + | |||
| + | ### | ||
| + | |||
| + | {{: | ||
| + | |||
| + | ### | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ Operator ^ Orientation ^ | ||
| + | ^ $\text{E}$ | $\{0,0,0\}$ , | | ||
| + | ^ $C_3$ | $\{0,0,1\}$ , $\{0, | ||
| + | ^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3}, | ||
| + | ^ $\text{i}$ | $\{0,0,0\}$ , | | ||
| + | ^ $S_6$ | $\{0,0,1\}$ , $\{0, | ||
| + | ^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3}, | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Different Settings ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Character Table ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ \text{E} \, | ||
| + | ^ $ A_{1g} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | | ||
| + | ^ $ A_{2g} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | | ||
| + | ^ $ E_g $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 2 $ | $ -1 $ | $ 0 $ | | ||
| + | ^ $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | | ||
| + | ^ $ A_{2u} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ 1 $ | | ||
| + | ^ $ E_u $ | $ 2 $ | $ -1 $ | $ 0 $ | $ -2 $ | $ 1 $ | $ 0 $ | | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Product Table ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ A_{1g} $ ^ $ A_{2g} $ ^ $ E_g $ ^ $ A_{1u} $ ^ $ A_{2u} $ ^ $ E_u $ ^ | ||
| + | ^ $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | | ||
| + | ^ $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | | ||
| + | ^ $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | | ||
| + | ^ $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | | ||
| + | ^ $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | | ||
| + | ^ $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Sub Groups with compatible settings ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Super Groups with compatible settings ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | * [[physics_chemistry: | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Invariant Potential expanded on renormalized spherical Harmonics ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | Any potential (function) can be written as a sum over spherical harmonics. | ||
| + | $$V(r, | ||
| + | Here $A_{k, | ||
| + | The presence of symmetry induces relations between the expansion coefficients such that $V(r, | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Expansion ==== | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Input format suitable for Mathematica (Quanty.nb) | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty | ||
| + | |||
| + | Akm[k_, | ||
| </ | </ | ||
| - | ==== Result ==== | + | ### |
| - | <WRAP center box 100%> | + | |
| - | text produced as output | + | |
| - | </ | + | |
| - | ===== Table of contents | + | ==== Input format suitable for Quanty |
| - | {{indexmenu> | + | |
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, A(0,0)} , | ||
| + | {2, 0, A(2,0)} , | ||
| + | {4, 0, A(4,0)} , | ||
| + | | ||
| + | {4, 3, A(4,3)} , | ||
| + | {6, 0, A(6,0)} , | ||
| + | | ||
| + | {6, 3, A(6,3)} , | ||
| + | | ||
| + | {6, 6, A(6,6)} } | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== One particle coupling on a basis of spherical harmonics ==== | ||
| + | |||
| + | ### | ||
| + | |||
| + | The operator representing the potential in second quantisation is given as: | ||
| + | $$ O = \sum_{n'', | ||
| + | For the quantisation of the wave-function (physical meaning of the indices n,l,m) we can choose a basis of spherical harmonics times some radial function, i.e. $\psi_{n, | ||
| + | $$ A_{n'' | ||
| + | Note the difference between the function $A_{k,m}$ and the parameter $A_{n'' | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | we can express the operator as | ||
| + | $$ O = \sum_{n'', | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | The table below shows the expectation value of $O$ on a basis of spherical harmonics. We suppressed the principle quantum number indices. Note that in principle $A_{l'', | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ | ||
| + | ^$ {Y_{0}^{(0)}} $|$ \text{Ass}(0, | ||
| + | ^$ {Y_{-1}^{(1)}} $|$\color{darkred}{ 0 }$|$ \text{App}(0, | ||
| + | ^$ {Y_{0}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0, | ||
| + | ^$ {Y_{1}^{(1)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0, | ||
| + | ^$ {Y_{-2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0, | ||
| + | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2, | ||
| + | ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4, | ||
| + | ^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{5}{7}} \text{Add}(4, | ||
| + | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{\text{Apf}(4, | ||
| + | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{1}{3} \text{Apf}(4, | ||
| + | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2, | ||
| + | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{3}{5} \sqrt{\frac{3}{7}} \text{Apf}(2, | ||
| + | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2, | ||
| + | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \text{Apf}(4, | ||
| + | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\text{Apf}(4, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Rotation matrix to symmetry adapted functions (choice is not unique) ==== | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | Instead of a basis of spherical harmonics one can chose any other basis, which is given by a unitary transformation. Here we choose a rotation that simplifies the representation of the crystal field | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ | ||
| + | ^$ \text{s} $|$ 1 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
| + | ^$ p_x $|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ p_y $|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 1 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$ 0 $|$ -\frac{i}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
| + | ^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \frac{1}{\sqrt{6}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
| + | ^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $|$ 0 $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
| + | ^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
| + | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
| + | ^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{2}}{3} $| | ||
| + | ^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|$ \frac{i}{2 \sqrt{3}} $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $| | ||
| + | ^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{\sqrt{\frac{5}{2}}}{3} $|$ 0 $|$ 0 $|$ -\frac{2}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{\frac{5}{2}}}{3} $| | ||
| + | ^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | ==== One particle coupling on a basis of symmetry adapted functions ==== | ||
| + | |||
| + | ### | ||
| + | |||
| + | After rotation we find | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{\text{xy}-\sqrt{2}\text{yz}} $ ^ $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $ ^ $ d_{\text{yz}+\sqrt{2}\text{xy}} $ ^ $ d_{x^2-y^2-\sqrt{2}\text{xz}} $ ^ $ d_{3z^2-r^2} $ ^ $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $ ^ $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $ ^ $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $ ^ | ||
| + | ^$ \text{s} $|$ \text{Ass}(0, | ||
| + | ^$ p_x $|$\color{darkred}{ 0 }$|$ \text{App}(0, | ||
| + | ^$ p_y $|$\color{darkred}{ 0 }$|$ 0 $|$ \text{App}(0, | ||
| + | ^$ p_z $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0, | ||
| + | ^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0, | ||
| + | ^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{7} \sqrt{2} \text{Add}(2, | ||
| + | ^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{1}{7} \sqrt{2} \text{Add}(2, | ||
| + | ^$ d_{3z^2-r^2} $|$ \frac{\text{Asd}(2, | ||
| + | ^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \sqrt{\frac{3}{35}} \text{Apf}(2, | ||
| + | ^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2, | ||
| + | ^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{5} \sqrt{\frac{3}{7}} \text{Apf}(2, | ||
| + | ^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{2}{5} \sqrt{\frac{3}{7}} \text{Apf}(2, | ||
| + | ^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$\color{darkred}{ 0 }$|$ -\sqrt{\frac{3}{35}} \text{Apf}(2, | ||
| + | ^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ -\sqrt{\frac{3}{35}} \text{Apf}(2, | ||
| + | ^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Coupling for a single shell ===== | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | Although the parameters $A_{l'', | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | Click on one of the subsections to expand it or < | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Potential for s orbitals ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | 0 & \text{True} | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, Ea1g} } | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ | ||
| + | ^$ {Y_{0}^{(0)}} $|$ \text{Ea1g} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ \text{s} $ ^ | ||
| + | ^$ \text{s} $|$ \text{Ea1g} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Rotation matrix used** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ | ||
| + | ^$ \text{s} $|$ 1 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Ea1g}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ==== Potential for p orbitals ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, (1/3)*(Ea2u + (2)*(Eeu))} , | ||
| + | {2, 0, (5/3)*(Ea2u + (-1)*(Eeu))} } | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
| + | ^$ {Y_{-1}^{(1)}} $|$ \text{Eeu} $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea2u} $|$ 0 $| | ||
| + | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Eeu} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ | ||
| + | ^$ p_x $|$ \text{Eeu} $|$ 0 $|$ 0 $| | ||
| + | ^$ p_y $|$ 0 $|$ \text{Eeu} $|$ 0 $| | ||
| + | ^$ p_z $|$ 0 $|$ 0 $|$ \text{Ea2u} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Rotation matrix used** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
| + | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
| + | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
| + | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Eeu}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeu}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Ea2u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ==== Potential for d orbitals ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, (1/5)*(Ea1g + (2)*(EegPi + EegSigma))} , | ||
| + | {2, 0, Ea1g + (-1)*(EegPi) + (-2)*((sqrt(2))*(Meg))} , | ||
| + | {4, 0, (1/ | ||
| + | | ||
| + | {4, 3, (sqrt(7/ | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
| + | ^$ {Y_{-2}^{(2)}} $|$ \frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(\sqrt{2} \text{Eeg$\pi $}-\sqrt{2} \text{Eeg$\sigma $}-\text{Meg}\right) $|$ 0 $| | ||
| + | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(-\sqrt{2} \text{Eeg$\pi $}+\sqrt{2} \text{Eeg$\sigma $}+\text{Meg}\right) $| | ||
| + | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea1g} $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{1}^{(2)}} $|$ \frac{1}{3} \left(\sqrt{2} \text{Eeg$\pi $}-\sqrt{2} \text{Eeg$\sigma $}-\text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(\text{Eeg$\pi $}+2 \text{Eeg$\sigma $}-2 \sqrt{2} \text{Meg}\right) $|$ 0 $| | ||
| + | ^$ {Y_{2}^{(2)}} $|$ 0 $|$ \frac{1}{3} \left(-\sqrt{2} \text{Eeg$\pi $}+\sqrt{2} \text{Eeg$\sigma $}+\text{Meg}\right) $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(2 \text{Eeg$\pi $}+\text{Eeg$\sigma $}+2 \sqrt{2} \text{Meg}\right) $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ d_{\text{xy}-\sqrt{2}\text{yz}} $ ^ $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $ ^ $ d_{\text{yz}+\sqrt{2}\text{xy}} $ ^ $ d_{x^2-y^2-\sqrt{2}\text{xz}} $ ^ $ d_{3z^2-r^2} $ ^ | ||
| + | ^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ 0 $| | ||
| + | ^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ 0 $|$ \text{Eeg$\sigma $} $|$ 0 $|$ \text{Meg} $|$ 0 $| | ||
| + | ^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $|$ 0 $| | ||
| + | ^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ 0 $|$ \text{Meg} $|$ 0 $|$ \text{Eeg$\pi $} $|$ 0 $| | ||
| + | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1g} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Rotation matrix used** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
| + | ^$ d_{\text{xy}-\sqrt{2}\text{yz}} $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $|$ 0 $|$ -\frac{i}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $| | ||
| + | ^$ d_{x^2-y^2+2\sqrt{2}\text{xz}} $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \frac{1}{\sqrt{6}} $| | ||
| + | ^$ d_{\text{yz}+\sqrt{2}\text{xy}} $|$ \frac{i}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $|$ 0 $|$ \frac{i}{\sqrt{6}} $|$ -\frac{i}{\sqrt{3}} $| | ||
| + | ^$ d_{x^2-y^2-\sqrt{2}\text{xz}} $|$ \frac{1}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $| | ||
| + | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Eeg$\sigma $}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeg$\sigma $}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeg$\pi $}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeg$\pi $}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Ea1g}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ==== Potential for f orbitals ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, (1/7)*(Ea1u + Ea2u1 + Ea2u2 + (2)*(Eeu1) + (2)*(Eeu2))} , | ||
| + | {2, 0, (-5/ | ||
| + | {4, 0, (1/ | ||
| + | | ||
| + | {4, 3, (1/ | ||
| + | {6, 0, (-13/ | ||
| + | {6, 3, (-13/ | ||
| + | | ||
| + | | ||
| + | {6, 6, (-13/ | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <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)}} $|$ \frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) $|$ 0 $|$ 0 $|$ \frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{18} \left(9 \text{Ea1u}-4 \left(\text{Ea2u1}+\sqrt{5} \text{Ma2u}\right)-5 \text{Ea2u2}\right) $| | ||
| + | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}+2 \sqrt{5} \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}+4 \text{Meu}\right) $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}-2 \sqrt{5} \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-4 \text{Meu}\right) $|$ 0 $| | ||
| + | ^$ {Y_{0}^{(3)}} $|$ \frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{9} \left(5 \text{Ea2u1}+4 \text{Ea2u2}-4 \sqrt{5} \text{Ma2u}\right) $|$ 0 $|$ 0 $|$ -\frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $| | ||
| + | ^$ {Y_{1}^{(3)}} $|$ 0 $|$ \frac{1}{6} \left(-\sqrt{5} \text{Eeu1}+\sqrt{5} \text{Eeu2}+4 \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(\text{Eeu1}+5 \text{Eeu2}-2 \sqrt{5} \text{Meu}\right) $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(\sqrt{5} \text{Eeu1}-\sqrt{5} \text{Eeu2}-4 \text{Meu}\right) $|$ 0 $|$ 0 $|$ \frac{1}{6} \left(5 \text{Eeu1}+\text{Eeu2}+2 \sqrt{5} \text{Meu}\right) $|$ 0 $| | ||
| + | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{18} \left(9 \text{Ea1u}-4 \left(\text{Ea2u1}+\sqrt{5} \text{Ma2u}\right)-5 \text{Ea2u2}\right) $|$ 0 $|$ 0 $|$ -\frac{2 \sqrt{5} \text{Ea2u1}-2 \sqrt{5} \text{Ea2u2}+\text{Ma2u}}{9 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{18} \left(9 \text{Ea1u}+4 \text{Ea2u1}+5 \text{Ea2u2}+4 \sqrt{5} \text{Ma2u}\right) $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $ ^ $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $ ^ $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $ ^ | ||
| + | ^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$ \text{Ea2u1} $|$ 0 $|$ 0 $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ \text{Eeu1} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $| | ||
| + | ^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$ \text{Ma2u} $|$ 0 $|$ 0 $|$ \text{Ea2u2} $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ \text{Meu} $|$ 0 $|$ 0 $|$ \text{Eeu2} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea1u} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Rotation matrix used** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ | ||
| + | ^$ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $|$ \frac{\sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{2}}{3} $| | ||
| + | ^$ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $|$ \frac{i}{2 \sqrt{3}} $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $| | ||
| + | ^$ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $|$ \frac{\sqrt{\frac{5}{2}}}{3} $|$ 0 $|$ 0 $|$ -\frac{2}{3} $|$ 0 $|$ 0 $|$ -\frac{\sqrt{\frac{5}{2}}}{3} $| | ||
| + | ^$ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{1}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Ea2u1}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeu1}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeu1}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Ea2u2}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeu2}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeu2}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Ea1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ===== Coupling between two shells ===== | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | Click on one of the subsections to expand it or < | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Potential for s-d orbital mixing ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | 0 & k\neq 2\lor m\neq 0 \\ | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{2, 0, (sqrt(5))*(Ma1g)} } | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <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)}} $|$ 0 $|$ 0 $|$ \text{Ma1g} $|$ 0 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ d_{\text{xy}-\sqrt{2}\text{yz}} $ ^ $ d_{x^2-y^2+2\sqrt{2}\text{xz}} $ ^ $ d_{\text{yz}+\sqrt{2}\text{xy}} $ ^ $ d_{x^2-y^2-\sqrt{2}\text{xz}} $ ^ $ d_{3z^2-r^2} $ ^ | ||
| + | ^$ \text{s} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ma1g} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ==== Potential for p-f orbital mixing ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | 0 & (k\neq 4\land (k\neq 2\lor m\neq 0))\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_D3d_Zy_A.Quanty> | ||
| + | |||
| + | Akm = {{2, 0, (-5)*((1/ | ||
| + | {4, 0, (1/ | ||
| + | {4, 3, (-1)*((sqrt(3/ | ||
| + | | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <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)}} $|$ 0 $|$ 0 $|$ \frac{-2 \sqrt{5} \text{Ma2u1}-5 \text{Ma2u2}+6 \text{Meu1}}{\sqrt{6}} $|$ 0 $|$ 0 $|$ \frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{\sqrt{6}} $|$ 0 $| | ||
| + | ^$ {Y_{0}^{(1)}} $|$ \frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{3 \sqrt{2}} $|$ 0 $|$ 0 $|$ \frac{1}{3} \left(\sqrt{5} \text{Ma2u1}-2 \text{Ma2u2}\right) $|$ 0 $|$ 0 $|$ -\frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{3 \sqrt{2}} $| | ||
| + | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ -\frac{2 \text{Ma2u1}+\sqrt{5} \text{Ma2u2}}{\sqrt{6}} $|$ 0 $|$ 0 $|$ \frac{-2 \sqrt{5} \text{Ma2u1}-5 \text{Ma2u2}+6 \text{Meu1}}{\sqrt{6}} $|$ 0 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ f_{\sqrt{2}\backslash x^3-\left.3\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2-3\backslash z+5\left\backslash z^3\right.} $ ^ $ f_{\left.5\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash z-\left.5\left\backslash \sqrt{2}\right.\backslash y^2\right\backslash z+x\left\backslash \left(-1+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-y\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z-5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{5\left\backslash \sqrt{2}\right.\backslash x^3-\left.15\left\backslash \sqrt{2}\right.\right\backslash x\backslash y^2+4\backslash z\left\backslash \left(3-5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{x+\left.\sqrt{2}\backslash x^2\right\backslash z-\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash x\left\backslash z^2\right.} $ ^ $ f_{-y\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash x\right\backslash z+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-y\left\backslash \left(-3\left\backslash x^2\right.+y^2\right)\right.} $ ^ | ||
| + | ^$ p_x $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ 2 \text{Ma2u1}+\sqrt{5} (\text{Ma2u2}-\text{Meu1}) $|$ 0 $|$ 0 $| | ||
| + | ^$ p_y $|$ 0 $|$ 0 $|$ \text{Meu1} $|$ 0 $|$ 0 $|$ 2 \text{Ma2u1}+\sqrt{5} (\text{Ma2u2}-\text{Meu1}) $|$ 0 $| | ||
| + | ^$ p_z $|$ \text{Ma2u1} $|$ 0 $|$ 0 $|$ \text{Ma2u2} $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ===== Table of several point groups ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | [[physics_chemistry: | ||
| + | |||
| + | ### | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^Nonaxial groups | ||
| + | ^C< | ||
| + | ^D< | ||
| + | ^C< | ||
| + | ^C< | ||
| + | ^D< | ||
| + | ^D< | ||
| + | ^S< | ||
| + | ^Cubic groups | [[physics_chemistry: | ||
| + | ^Linear groups | ||
| + | |||
| + | ### | ||
