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physics_chemistry:point_groups:c6:orientation_z [2018/03/21 15:13] – created Stefano Agrestini | physics_chemistry:point_groups:c6:orientation_z [2024/06/27 09:19] (current) – Typo: indices m and l were switched in spherical harmonics in "we can express the operator as ..." Finn Keuchel | ||
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- | ====== Orientation | + | ~~CLOSETOC~~ |
+ | |||
+ | ====== Orientation | ||
+ | |||
+ | ===== Symmetry Operations | ||
### | ### | ||
- | alligned paragraph text | + | |
+ | In the C6 Point Group, with orientation Z there are the following symmetry operations | ||
### | ### | ||
- | ===== Example ===== | + | ### |
+ | |||
+ | {{: | ||
### | ### | ||
- | description text | + | |
### | ### | ||
- | ==== Input ==== | + | ^ Operator ^ Orientation ^ |
- | <code Quanty | + | ^ $\text{E}$ | $\{0,0,0\}$ , | |
- | -- some example code | + | ^ $C_6$ | $\{0,0,1\}$ , $\{0, |
+ | ^ $C_3$ | $\{0,0,1\}$ , $\{0, | ||
+ | ^ $C_2$ | $\{0,0,1\}$ , | | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Different Settings ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Character Table ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ \text{E} \, | ||
+ | ^ $ \text{A} $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | | ||
+ | ^ $ \text{B} $ | $ 1 $ | $ -1 $ | $ 1 $ | $ -1 $ | | ||
+ | ^ $ E_1 $ | $ 2 $ | $ 1 $ | $ -1 $ | $ -2 $ | | ||
+ | ^ $ E_2 $ | $ 2 $ | $ -1 $ | $ -1 $ | $ 2 $ | | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Product Table ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ \text{A} $ ^ $ \text{B} $ ^ $ E_1 $ ^ $ E_2 $ ^ | ||
+ | ^ $ \text{A} $ | $ \text{A} $ | $ \text{B} $ | $ E_1 $ | $ E_2 $ | | ||
+ | ^ $ \text{B} $ | $ \text{B} $ | $ \text{A} $ | $ E_2 $ | $ E_1 $ | | ||
+ | ^ $ E_1 $ | $ E_1 $ | $ E_2 $ | $ 2 \text{A}+E_2 $ | $ 2 \text{B}+E_1 $ | | ||
+ | ^ $ E_2 $ | $ E_2 $ | $ E_1 $ | $ 2 \text{B}+E_1 $ | $ 2 \text{A}+E_2 $ | | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Sub Groups with compatible settings ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | * [[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_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, A(0,0)} , | ||
+ | {1, 0, A(1,0)} , | ||
+ | {2, 0, A(2,0)} , | ||
+ | {3, 0, A(3,0)} , | ||
+ | {4, 0, A(4,0)} , | ||
+ | {5, 0, A(5,0)} , | ||
+ | {6, 0, A(6,0)} , | ||
+ | | ||
+ | {6, 6, A(6,6) + (I)*(B(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}{ \frac{\text{Asp}(1, | ||
+ | ^$ {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}{ \frac{\text{Apd}(1, | ||
+ | ^$ {Y_{0}^{(2)}} $|$ \frac{\text{Asd}(2, | ||
+ | ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1, | ||
+ | ^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
+ | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Aff}(0, | ||
+ | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1, | ||
+ | ^$ {Y_{-1}^{(3)}} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2, | ||
+ | ^$ {Y_{0}^{(3)}} $|$\color{darkred}{ \frac{\text{Asf}(3, | ||
+ | ^$ {Y_{1}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2, | ||
+ | ^$ {Y_{2}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1, | ||
+ | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} (\text{Aff}(6, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | ==== 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_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 $| | ||
+ | ^$ 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 $| | ||
+ | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$\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 }$| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$\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} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$| | ||
+ | ^$ f_{y\left(3x^2-y^2\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}} $| | ||
+ | ^$ f_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ f_{y\left(5z^2-r^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 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(5z^2-3r^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 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(5z^2-r^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 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(x^2-y^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}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ f_{x\left(x^2-3y^2\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_y $ ^ $ p_z $ ^ $ p_x $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ | ||
+ | ^$ \text{s} $|$ \text{Ass}(0, | ||
+ | ^$ p_y $|$\color{darkred}{ 0 }$|$ \text{App}(0, | ||
+ | ^$ p_z $|$\color{darkred}{ \frac{\text{Asp}(1, | ||
+ | ^$ p_x $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{App}(0, | ||
+ | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0, | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ \frac{\text{Apd}(1, | ||
+ | ^$ d_{3z^2-r^2} $|$ \frac{\text{Asd}(2, | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Apd}(1, | ||
+ | ^$ d_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
+ | ^$ f_{y\left(3x^2-y^2\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_{\text{xyz}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ \frac{\text{Adf}(1, | ||
+ | ^$ f_{y\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2, | ||
+ | ^$ f_{z\left(5z^2-3r^2\right)} $|$\color{darkred}{ \frac{\text{Asf}(3, | ||
+ | ^$ f_{x\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{3}{5} \sqrt{\frac{2}{7}} \text{Apf}(2, | ||
+ | ^$ f_{z\left(x^2-y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{\text{Adf}(1, | ||
+ | ^$ f_{x\left(x^2-3y^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{10}{13} \sqrt{\frac{7}{33}} \text{Bff}(6, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | ===== 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_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, Ea} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ | ||
+ | ^$ {Y_{0}^{(0)}} $|$ \text{Ea} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ \text{s} $ ^ | ||
+ | ^$ \text{s} $|$ \text{Ea} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Rotation matrix used** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ | ||
+ | ^$ \text{s} $|$ 1 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ea}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for p orbitals ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | 0 & k\neq 2\lor m\neq 0 \\ | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, (1/3)*(Ea + (2)*(Ee1))} , | ||
+ | {2, 0, (5/3)*(Ea + (-1)*(Ee1))} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
+ | ^$ {Y_{-1}^{(1)}} $|$ \text{Ee1} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Ea} $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Ee1} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ | ||
+ | ^$ p_y $|$ \text{Ee1} $|$ 0 $|$ 0 $| | ||
+ | ^$ p_z $|$ 0 $|$ \text{Ea} $|$ 0 $| | ||
+ | ^$ p_x $|$ 0 $|$ 0 $|$ \text{Ee1} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Rotation matrix used** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
+ | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
+ | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| | ||
+ | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ee1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ea}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for d orbitals ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | 0 & (k\neq 2\land k\neq 4)\lor m\neq 0 \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, (1/5)*(Ea + (2)*(Ee1 + Ee2))} , | ||
+ | {2, 0, Ea + Ee1 + (-2)*(Ee2)} , | ||
+ | {4, 0, (3/ | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <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)}} $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| | ||
+ | ^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ | ||
+ | ^$ d_{\text{xy}} $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| | ||
+ | ^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Rotation matrix used** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
+ | ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ee2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ea}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for f orbitals ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | 0 & (k\neq 6\land ((k\neq 2\land k\neq 4)\lor m\neq 0))\lor (m\neq -6\land m\neq 0\land m\neq 6) \\ | ||
+ | | ||
+ | | ||
+ | | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, (1/7)*(Ea + Ebxx2y2 + Ebyx2y2 + (2)*(Ee1) + (2)*(Ee2))} , | ||
+ | {2, 0, (5/ | ||
+ | {4, 0, (3/ | ||
+ | {6, 0, (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{\text{Ebxx2y2}+\text{Ebyx2y2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{2} (-\text{Ebxx2y2}+\text{Ebyx2y2}-2 i \text{Mb}) $| | ||
+ | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| | ||
+ | ^$ {Y_{3}^{(3)}} $|$ \frac{1}{2} (-\text{Ebxx2y2}+\text{Ebyx2y2}+2 i \text{Mb}) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ebxx2y2}+\text{Ebyx2y2}}{2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ | ||
+ | ^$ f_{y\left(3x^2-y^2\right)} $|$ \text{Ebyx2y2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mb} $| | ||
+ | ^$ f_{\text{xyz}} $|$ 0 $|$ \text{Ee1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Ee2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(5z^2-3r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ea} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee2} $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ee1} $|$ 0 $| | ||
+ | ^$ f_{x\left(x^2-3y^2\right)} $|$ \text{Mb} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Ebxx2y2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <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_{y\left(3x^2-y^2\right)} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
+ | ^$ f_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ f_{y\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(5z^2-3r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ f_{x\left(x^2-3y^2\right)} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ebyx2y2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ea}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ebxx2y2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ===== Coupling between two shells ===== | ||
+ | |||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | Click on one of the subsections to expand it or < | ||
+ | |||
+ | ### | ||
+ | |||
+ | ==== Potential for s-p orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & k\neq 1\lor m\neq 0 \\ | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{1, 0, A(1,0)} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
+ | ^$ {Y_{0}^{(0)}} $|$ 0 $|$ \frac{A(1, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ p_y $ ^ $ p_z $ ^ $ p_x $ ^ | ||
+ | ^$ \text{s} $|$ 0 $|$ \frac{A(1, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== 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}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{2, 0, A(2,0)} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
+ | ^$ {Y_{0}^{(0)}} $|$ 0 $|$ 0 $|$ \frac{A(2, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ | ||
+ | ^$ \text{s} $|$ 0 $|$ 0 $|$ \frac{A(2, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for s-f orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & k\neq 3\lor m\neq 0 \\ | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{3, 0, A(3,0)} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ | ||
+ | ^$ {Y_{0}^{(0)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{A(3, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ | ||
+ | ^$ \text{s} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{A(3, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for p-d orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & (k\neq 1\land k\neq 3)\lor m\neq 0 \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{1, 0, A(1,0)} , | ||
+ | {3, 0, A(3,0)} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
+ | ^$ {Y_{-1}^{(1)}} $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ d_{\text{xy}} $ ^ $ d_{\text{yz}} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{xz}} $ ^ $ d_{x^2-y^2} $ ^ | ||
+ | ^$ p_y $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ p_z $|$ 0 $|$ 0 $|$ \frac{14 A(1,0)+9 A(3,0)}{7 \sqrt{15}} $|$ 0 $|$ 0 $| | ||
+ | ^$ p_x $|$ 0 $|$ 0 $|$ 0 $|$ \frac{7 A(1,0)-3 A(3,0)}{7 \sqrt{5}} $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for p-f orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & (k\neq 2\land k\neq 4)\lor m\neq 0 \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{2, 0, A(2,0)} , | ||
+ | {4, 0, A(4,0)} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <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{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ | ||
+ | ^$ p_y $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \frac{27 A(2,0)+20 A(4,0)}{15 \sqrt{21}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ p_x $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{15} \sqrt{\frac{2}{7}} (9 A(2,0)-5 A(4,0)) $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for d-f orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & (k\neq 1\land k\neq 3\land k\neq 5)\lor m\neq 0 \\ | ||
+ | | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_C6_Z.Quanty> | ||
+ | |||
+ | Akm = {{1, 0, A(1,0)} , | ||
+ | {3, 0, A(3,0)} , | ||
+ | {5, 0, A(5,0)} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-3}^{(3)}} $ ^ $ {Y_{-2}^{(3)}} $ ^ $ {Y_{-1}^{(3)}} $ ^ $ {Y_{0}^{(3)}} $ ^ $ {Y_{1}^{(3)}} $ ^ $ {Y_{2}^{(3)}} $ ^ $ {Y_{3}^{(3)}} $ ^ | ||
+ | ^$ {Y_{-2}^{(2)}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{y\left(3x^2-y^2\right)} $ ^ $ f_{\text{xyz}} $ ^ $ f_{y\left(5z^2-r^2\right)} $ ^ $ f_{z\left(5z^2-3r^2\right)} $ ^ $ f_{x\left(5z^2-r^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ $ f_{x\left(x^2-3y^2\right)} $ ^ | ||
+ | ^$ d_{\text{xy}} $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{99 A(1,0)+44 A(3,0)+50 A(5,0)}{33 \sqrt{35}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{33} \sqrt{\frac{2}{35}} (66 A(1,0)+11 A(3,0)-25 A(5,0)) $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \frac{33 A(1,0)-22 A(3,0)+5 A(5,0)}{33 \sqrt{7}} $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ===== Table of several point groups ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | [[physics_chemistry: | ||
+ | |||
+ | ### | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^Nonaxial groups | ||
+ | ^C< | ||
+ | ^D< | ||
+ | ^C< | ||
+ | ^C< | ||
+ | ^D< | ||
+ | ^D< | ||
+ | ^S< | ||
+ | ^Cubic groups | [[physics_chemistry: | ||
+ | ^Linear groups | ||
+ | |||
+ | ### |