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| physics_chemistry:point_groups:oh:orientation_0sqrt21z [2018/03/21 18:49] – created Stefano Agrestini | physics_chemistry:point_groups:oh:orientation_0sqrt21z [2018/09/06 12:50] (current) – Maurits W. Haverkort | ||
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| + | ~~CLOSETOC~~ | ||
| + | |||
| ====== Orientation 0sqrt21z ====== | ====== Orientation 0sqrt21z ====== | ||
| + | |||
| + | ===== Symmetry Operations ===== | ||
| ### | ### | ||
| - | alligned paragraph text | + | |
| + | In the Oh Point Group, with orientation 0sqrt21z there are the following symmetry operations | ||
| ### | ### | ||
| - | ===== Example ===== | + | ### |
| + | |||
| + | {{: | ||
| ### | ### | ||
| - | description text | + | |
| ### | ### | ||
| - | ==== Input ==== | + | ^ Operator ^ Orientation ^ |
| - | <code Quanty | + | ^ $\text{E}$ | $\{0,0,0\}$ , | |
| - | -- some example code | + | ^ $C_3$ | $\{0,0,1\}$ , $\{0, |
| + | ^ $C_2$ | $\{1,0,0\}$ , $\left\{1, | ||
| + | ^ $C_4$ | $\left\{0, | ||
| + | ^ $C_2$ | $\left\{0, | ||
| + | ^ $\text{i}$ | $\{0,0,0\}$ , | | ||
| + | ^ $S_4$ | $\left\{0, | ||
| + | ^ $S_6$ | $\{0,0,1\}$ , $\{0, | ||
| + | ^ $\sigma _h$ | $\left\{0, | ||
| + | ^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1, | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Different Settings ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | * [[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 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | | ||
| + | ^ $ A_{2g} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ 1 $ | $ -1 $ | | ||
| + | ^ $ E_g $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 0 $ | $ 2 $ | $ 2 $ | $ 0 $ | $ -1 $ | $ 2 $ | $ 0 $ | | ||
| + | ^ $ T_{1g} $ | $ 3 $ | $ 0 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 3 $ | $ 1 $ | $ 0 $ | $ -1 $ | $ -1 $ | | ||
| + | ^ $ T_{2g} $ | $ 3 $ | $ 0 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 3 $ | $ -1 $ | $ 0 $ | $ -1 $ | $ 1 $ | | ||
| + | ^ $ A_{1u} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | $ -1 $ | | ||
| + | ^ $ A_{2u} $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ 1 $ | | ||
| + | ^ $ E_u $ | $ 2 $ | $ -1 $ | $ 0 $ | $ 0 $ | $ 2 $ | $ -2 $ | $ 0 $ | $ 1 $ | $ -2 $ | $ 0 $ | | ||
| + | ^ $ T_{1u} $ | $ 3 $ | $ 0 $ | $ -1 $ | $ 1 $ | $ -1 $ | $ -3 $ | $ -1 $ | $ 0 $ | $ 1 $ | $ 1 $ | | ||
| + | ^ $ T_{2u} $ | $ 3 $ | $ 0 $ | $ 1 $ | $ -1 $ | $ -1 $ | $ -3 $ | $ 1 $ | $ 0 $ | $ 1 $ | $ -1 $ | | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Product Table ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ A_{1g} $ ^ $ A_{2g} $ ^ $ E_g $ ^ $ T_{1g} $ ^ $ T_{2g} $ ^ $ A_{1u} $ ^ $ A_{2u} $ ^ $ E_u $ ^ $ T_{1u} $ ^ $ T_{2u} $ ^ | ||
| + | ^ $ A_{1g} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ T_{1g} $ | $ T_{2g} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ T_{1u} $ | $ T_{2u} $ | | ||
| + | ^ $ A_{2g} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ T_{2g} $ | $ T_{1g} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ T_{2u} $ | $ T_{1u} $ | | ||
| + | ^ $ E_g $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ T_{1g}+T_{2g} $ | $ T_{1g}+T_{2g} $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ T_{1u}+T_{2u} $ | $ T_{1u}+T_{2u} $ | | ||
| + | ^ $ T_{1g} $ | $ T_{1g} $ | $ T_{2g} $ | $ T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | $ T_{1u} $ | $ T_{2u} $ | $ T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | | ||
| + | ^ $ T_{2g} $ | $ T_{2g} $ | $ T_{1g} $ | $ T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | $ T_{2u} $ | $ T_{1u} $ | $ T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | | ||
| + | ^ $ A_{1u} $ | $ A_{1u} $ | $ A_{2u} $ | $ E_u $ | $ T_{1u} $ | $ T_{2u} $ | $ A_{1g} $ | $ A_{2g} $ | $ E_g $ | $ T_{1g} $ | $ T_{2g} $ | | ||
| + | ^ $ A_{2u} $ | $ A_{2u} $ | $ A_{1u} $ | $ E_u $ | $ T_{2u} $ | $ T_{1u} $ | $ A_{2g} $ | $ A_{1g} $ | $ E_g $ | $ T_{2g} $ | $ T_{1g} $ | | ||
| + | ^ $ E_u $ | $ E_u $ | $ E_u $ | $ A_{1u}+A_{2u}+E_u $ | $ T_{1u}+T_{2u} $ | $ T_{1u}+T_{2u} $ | $ E_g $ | $ E_g $ | $ A_{1g}+A_{2g}+E_g $ | $ T_{1g}+T_{2g} $ | $ T_{1g}+T_{2g} $ | | ||
| + | ^ $ T_{1u} $ | $ T_{1u} $ | $ T_{2u} $ | $ T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | $ T_{1g} $ | $ T_{2g} $ | $ T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | | ||
| + | ^ $ T_{2u} $ | $ T_{2u} $ | $ T_{1u} $ | $ T_{1u}+T_{2u} $ | $ A_{2u}+E_u+T_{1u}+T_{2u} $ | $ A_{1u}+E_u+T_{1u}+T_{2u} $ | $ T_{2g} $ | $ T_{1g} $ | $ T_{1g}+T_{2g} $ | $ A_{2g}+E_g+T_{1g}+T_{2g} $ | $ A_{1g}+E_g+T_{1g}+T_{2g} $ | | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== 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: | ||
| + | * [[physics_chemistry: | ||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Super Groups with compatible settings ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | ===== Invariant Potential expanded on renormalized spherical Harmonics ===== | ||
| + | |||
| + | ### | ||
| + | |||
| + | Any potential (function) can be written as a sum over spherical harmonics. | ||
| + | $$V(r, | ||
| + | Here $A_{k, | ||
| + | The presence of symmetry induces relations between the expansion coefficients such that $V(r, | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Expansion ==== | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | i \sqrt{\frac{10}{7}} A(4,0) & k=4\land (m=-3\lor m=3) \\ | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== 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_Oh_0sqrt21z.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, A(0,0)} , | ||
| + | {4, 0, A(4,0)} , | ||
| + | | ||
| + | {4, 3, (I)*((sqrt(10/ | ||
| + | {6, 0, A(6,0)} , | ||
| + | | ||
| + | {6, 3, (-1/ | ||
| + | | ||
| + | {6, 6, (-1/ | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== 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)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{5}{21} i \sqrt{2} \text{Add}(4, | ||
| + | ^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ -\frac{5}{21} i \sqrt{2} \text{Add}(4, | ||
| + | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{10}{21}} \text{Apf}(4, | ||
| + | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4, | ||
| + | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4, | ||
| + | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{4 \text{Apf}(4, | ||
| + | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{2}{7}} \text{Apf}(4, | ||
| + | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} i \sqrt{\frac{10}{7}} \text{Apf}(4, | ||
| + | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{10}{21}} \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_{\sqrt{2}\text{xz}-\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\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_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{\sqrt{6}} $|$ \frac{i}{\sqrt{3}} $|$ 0 $|$ \frac{i}{\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{xz}-\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\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_{y^2-x^2-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{1}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|$ 0 $|$ -\frac{i}{\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_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\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{i \sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{i \sqrt{2}}{3} $| | ||
| + | ^$ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\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{1}{2 \sqrt{3}} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $| | ||
| + | ^$ f_{y+\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-5\backslash y\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{\sqrt{\frac{5}{3}}}{2} $|$ -\frac{i}{2 \sqrt{3}} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $| | ||
| + | ^$ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+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{1}{3} i \sqrt{\frac{5}{2}} $|$ 0 $|$ 0 $|$ \frac{2}{3} $|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{5}{2}} $| | ||
| + | ^$ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\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{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\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{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{x\left\backslash \left(x^2-3\left\backslash y^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{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | ==== One particle coupling on a basis of symmetry adapted functions ==== | ||
| + | |||
| + | ### | ||
| + | |||
| + | After rotation we find | ||
| + | |||
| + | ### | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{\sqrt{2}\text{xz}-\text{xy}} $ ^ $ d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)} $ ^ $ d_{\text{xz}-\sqrt{2}\text{xy}} $ ^ $ d_{y^2-x^2-\sqrt{2}\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.} $ ^ $ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.} $ ^ $ f_{y+\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-5\backslash y\left\backslash z^2\right.} $ ^ $ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.} $ ^ $ f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\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_{\sqrt{2}\text{xz}-\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0, | ||
| + | ^$ d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ d_{\text{xz}-\sqrt{2}\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ d_{y^2-x^2-\sqrt{2}\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
| + | ^$ f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\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 }$|$ \text{Aff}(0, | ||
| + | ^$ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.} $|$\color{darkred}{ 0 }$|$ \frac{2 \text{Apf}(4, | ||
| + | ^$ f_{y+\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-5\backslash y\left\backslash z^2\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{2 \text{Apf}(4, | ||
| + | ^$ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{2 \text{Apf}(4, | ||
| + | ^$ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\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 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0, | ||
| + | ^$ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\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 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Aff}(0, | ||
| + | ^$ f_{x\left\backslash \left(x^2-3\left\backslash y^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 $|$ 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_Oh_0sqrt21z.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.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** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | 0 & \text{True} | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, Et1u} } | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
| + | ^$ {Y_{-1}^{(1)}} $|$ \text{Et1u} $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Et1u} $|$ 0 $| | ||
| + | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Et1u} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ | ||
| + | ^$ p_x $|$ \text{Et1u} $|$ 0 $|$ 0 $| | ||
| + | ^$ p_y $|$ 0 $|$ \text{Et1u} $|$ 0 $| | ||
| + | ^$ p_z $|$ 0 $|$ 0 $|$ \text{Et1u} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Rotation matrix used** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
| + | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
| + | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
| + | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Et1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ==== Potential for d orbitals ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | -i \sqrt{\frac{14}{5}} (\text{Eeg}-\text{Et2g}) & k=4\land (m=-3\lor m=3) \\ | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, (1/ | ||
| + | {4, 0, (-7/5)*(Eeg + (-1)*(Et2g))} , | ||
| + | | ||
| + | {4, 3, (-I)*((sqrt(14/ | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <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} (\text{Eeg}+2 \text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $| | ||
| + | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $|$ 0 $|$ -\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $| | ||
| + | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{1}^{(2)}} $|$ -\frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (2 \text{Eeg}+\text{Et2g}) $|$ 0 $| | ||
| + | ^$ {Y_{2}^{(2)}} $|$ 0 $|$ \frac{1}{3} i \sqrt{2} (\text{Eeg}-\text{Et2g}) $|$ 0 $|$ 0 $|$ \frac{1}{3} (\text{Eeg}+2 \text{Et2g}) $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ d_{\sqrt{2}\text{xz}-\text{xy}} $ ^ $ d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)} $ ^ $ d_{\text{xz}-\sqrt{2}\text{xy}} $ ^ $ d_{y^2-x^2-\sqrt{2}\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ | ||
| + | ^$ d_{\sqrt{2}\text{xz}-\text{xy}} $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)} $|$ 0 $|$ \text{Eeg} $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ d_{\text{xz}-\sqrt{2}\text{xy}} $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $|$ 0 $| | ||
| + | ^$ d_{y^2-x^2-\sqrt{2}\text{yz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $|$ 0 $| | ||
| + | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2g} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Rotation matrix used** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
| + | ^$ d_{\sqrt{2}\text{xz}-\text{xy}} $|$ -\frac{i}{\sqrt{6}} $|$ \frac{1}{\sqrt{3}} $|$ 0 $|$ -\frac{1}{\sqrt{3}} $|$ \frac{i}{\sqrt{6}} $| | ||
| + | ^$ d_{\left.y^2-x^2+2\sqrt{2}\text{yz}\right)} $|$ -\frac{1}{\sqrt{6}} $|$ \frac{i}{\sqrt{3}} $|$ 0 $|$ \frac{i}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $| | ||
| + | ^$ d_{\text{xz}-\sqrt{2}\text{xy}} $|$ -\frac{i}{\sqrt{3}} $|$ -\frac{1}{\sqrt{6}} $|$ 0 $|$ \frac{1}{\sqrt{6}} $|$ \frac{i}{\sqrt{3}} $| | ||
| + | ^$ d_{y^2-x^2-\sqrt{2}\text{yz}} $|$ -\frac{1}{\sqrt{3}} $|$ -\frac{i}{\sqrt{6}} $|$ 0 $|$ -\frac{i}{\sqrt{6}} $|$ -\frac{1}{\sqrt{3}} $| | ||
| + | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Eeg}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Eeg}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et2g}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et2g}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et2g}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ==== Potential for f orbitals ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | | ||
| + | i \sqrt{\frac{5}{14}} (2 \text{Ea2u}-3 \text{Et1u}+\text{Et2u}) & k=4\land (m=-3\lor m=3) \\ | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty> | ||
| + | |||
| + | Akm = {{0, 0, (1/7)*(Ea2u + (3)*(Et1u + Et2u))} , | ||
| + | {4, 0, (1/ | ||
| + | | ||
| + | {4, 3, (I)*((sqrt(5/ | ||
| + | {6, 0, (26/ | ||
| + | | ||
| + | {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} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u}) $|$ 0 $|$ 0 $|$ -\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) $| | ||
| + | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $|$ -\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $| | ||
| + | ^$ {Y_{0}^{(3)}} $|$ \frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} (5 \text{Ea2u}+4 \text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $| | ||
| + | ^$ {Y_{1}^{(3)}} $|$ 0 $|$ -\frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (\text{Et1u}+5 \text{Et2u}) $|$ 0 $|$ 0 $| | ||
| + | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{6} i \sqrt{5} (\text{Et1u}-\text{Et2u}) $|$ 0 $|$ 0 $|$ \frac{1}{6} (5 \text{Et1u}+\text{Et2u}) $|$ 0 $| | ||
| + | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}-9 \text{Et2u}) $|$ 0 $|$ 0 $|$ -\frac{1}{9} i \sqrt{10} (\text{Ea2u}-\text{Et1u}) $|$ 0 $|$ 0 $|$ \frac{1}{18} (4 \text{Ea2u}+5 \text{Et1u}+9 \text{Et2u}) $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.} $ ^ $ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.} $ ^ $ f_{y+\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-5\backslash y\left\backslash z^2\right.} $ ^ $ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.} $ ^ $ f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.} $ ^ | ||
| + | ^$ f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.} $|$ \text{Ea2u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.} $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{y+\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-5\backslash y\left\backslash z^2\right.} $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1u} $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $|$ 0 $| | ||
| + | ^$ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $|$ 0 $| | ||
| + | ^$ f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2u} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <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_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.} $|$ -\frac{i \sqrt{2}}{3} $|$ 0 $|$ 0 $|$ \frac{\sqrt{5}}{3} $|$ 0 $|$ 0 $|$ -\frac{i \sqrt{2}}{3} $| | ||
| + | ^$ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.} $|$ 0 $|$ \frac{1}{2} i \sqrt{\frac{5}{3}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $|$ \frac{1}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $| | ||
| + | ^$ f_{y+\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-5\backslash y\left\backslash z^2\right.} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ -\frac{i}{2 \sqrt{3}} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $| | ||
| + | ^$ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.} $|$ \frac{1}{3} i \sqrt{\frac{5}{2}} $|$ 0 $|$ 0 $|$ \frac{2}{3} $|$ 0 $|$ 0 $|$ \frac{1}{3} i \sqrt{\frac{5}{2}} $| | ||
| + | ^$ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.} $|$ 0 $|$ -\frac{i}{2 \sqrt{3}} $|$ -\frac{\sqrt{\frac{5}{3}}}{2} $|$ 0 $|$ \frac{\sqrt{\frac{5}{3}}}{2} $|$ \frac{i}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.} $|$ 0 $|$ -\frac{1}{2 \sqrt{3}} $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{3}} $|$ -\frac{1}{2 \sqrt{3}} $|$ 0 $| | ||
| + | ^$ f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Irriducible representations and their onsite energy** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | ^ ^$$\text{Ea2u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et1u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et2u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et2u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | ^ ^$$\text{Et2u}$$ | {{: | ||
| + | |$$\psi(\theta, | ||
| + | |$$\psi(\hat{x}, | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | ===== Coupling between two shells ===== | ||
| + | |||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | Click on one of the subsections to expand it or < | ||
| + | |||
| + | ### | ||
| + | |||
| + | ==== Potential for p-f orbital mixing ==== | ||
| + | |||
| + | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | ||
| + | 0 & k\neq 4\lor (m\neq -3\land m\neq 0\land m\neq 3) \\ | ||
| + | i \sqrt{\frac{15}{2}} \text{Mt1u} & k=4\land (m=-3\lor m=3) \\ | ||
| + | | ||
| + | \end{cases}$$ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty.nb> | ||
| + | |||
| + | Akm[k_, | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | <code Quanty Akm_Oh_0sqrt21z.Quanty> | ||
| + | |||
| + | Akm = {{4, 0, (1/ | ||
| + | | ||
| + | {4, 3, (I)*((sqrt(15/ | ||
| + | |||
| + | </ | ||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <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{\text{Mt1u}}{\sqrt{6}} $|$ 0 $|$ 0 $|$ i \sqrt{\frac{5}{6}} \text{Mt1u} $|$ 0 $| | ||
| + | ^$ {Y_{0}^{(1)}} $|$ -\frac{1}{3} i \sqrt{\frac{5}{2}} \text{Mt1u} $|$ 0 $|$ 0 $|$ \frac{2 \text{Mt1u}}{3} $|$ 0 $|$ 0 $|$ -\frac{1}{3} i \sqrt{\frac{5}{2}} \text{Mt1u} $| | ||
| + | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ i \sqrt{\frac{5}{6}} \text{Mt1u} $|$ 0 $|$ 0 $|$ -\frac{\text{Mt1u}}{\sqrt{6}} $|$ 0 $|$ 0 $| | ||
| + | |||
| + | |||
| + | ### | ||
| + | |||
| + | </ | ||
| + | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
| + | |||
| + | ### | ||
| + | |||
| + | | $ $ ^ $ f_{\left.-3\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y+\sqrt{2}\backslash y^3-3\backslash z+5\left\backslash z^3\right.} $ ^ $ f_{\left\backslash x\left\backslash \left(1+\left.\left.10\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z-5\left\backslash z^2\right.\right)\right.\right.} $ ^ $ f_{y+\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-5\backslash y\left\backslash z^2\right.} $ ^ $ f_{\left.15\left\backslash \sqrt{2}\right.\backslash x^2\right\backslash y-5\left\backslash \sqrt{2}\right.\backslash y^3+4\backslash z\left\backslash \left(-3+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{-x\left\backslash \left(-1+\left.\left.2\left\backslash \sqrt{2}\right.\right\backslash y\right\backslash z+5\left\backslash z^2\right.\right)\right.} $ ^ $ f_{y-\left.\sqrt{2}\backslash x^2\right\backslash z+\left.\sqrt{2}\backslash y^2\right\backslash z-5\backslash y\left\backslash z^2\right.} $ ^ $ f_{x\left\backslash \left(x^2-3\left\backslash y^2\right.\right)\right.} $ ^ | ||
| + | ^$ p_x $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ p_y $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
| + | ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt1u} $|$ 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 | ||
| + | |||
| + | ### | ||
