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physics_chemistry:point_groups:td:orientation_xyz [2018/03/21 18:45] – created Stefano Agrestini | physics_chemistry:point_groups:td:orientation_xyz [2018/04/06 08:56] (current) – Maurits W. Haverkort | ||
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+ | ~~CLOSETOC~~ | ||
+ | |||
====== Orientation xyz ====== | ====== Orientation xyz ====== | ||
+ | |||
+ | ===== Symmetry Operations ===== | ||
### | ### | ||
- | alligned paragraph text | + | |
+ | In the Td Point Group, with orientation xyz there are the following symmetry operations | ||
### | ### | ||
- | ===== Example ===== | + | ### |
+ | |||
+ | {{: | ||
### | ### | ||
- | description text | + | |
### | ### | ||
- | ==== Input ==== | + | ^ Operator ^ Orientation ^ |
- | <code Quanty | + | ^ $\text{E}$ | $\{0,0,0\}$ , | |
- | -- some example code | + | ^ $C_3$ | $\{1,1,1\}$ , $\{1, |
+ | ^ $C_2$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , | | ||
+ | ^ $S_4$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0, | ||
+ | ^ $\sigma _d$ | $\{1,1,0\}$ , $\{1, | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Different Settings ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | * [[physics_chemistry: | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Character Table ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ \text{E} \, | ||
+ | ^ $ A_1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ 1 $ | | ||
+ | ^ $ A_2 $ | $ 1 $ | $ 1 $ | $ 1 $ | $ -1 $ | $ -1 $ | | ||
+ | ^ $ \text{E} $ | $ 2 $ | $ -1 $ | $ 2 $ | $ 0 $ | $ 0 $ | | ||
+ | ^ $ T_1 $ | $ 3 $ | $ 0 $ | $ -1 $ | $ 1 $ | $ -1 $ | | ||
+ | ^ $ T_2 $ | $ 3 $ | $ 0 $ | $ -1 $ | $ -1 $ | $ 1 $ | | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Product Table ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ A_1 $ ^ $ A_2 $ ^ $ \text{E} $ ^ $ T_1 $ ^ $ T_2 $ ^ | ||
+ | ^ $ A_1 $ | $ A_1 $ | $ A_2 $ | $ \text{E} $ | $ T_1 $ | $ T_2 $ | | ||
+ | ^ $ A_2 $ | $ A_2 $ | $ A_1 $ | $ \text{E} $ | $ T_2 $ | $ T_1 $ | | ||
+ | ^ $ \text{E} $ | $ \text{E} $ | $ \text{E} $ | $ \text{E}+A_1+A_2 $ | $ T_1+T_2 $ | $ T_1+T_2 $ | | ||
+ | ^ $ T_1 $ | $ T_1 $ | $ T_2 $ | $ T_1+T_2 $ | $ \text{E}+A_1+T_1+T_2 $ | $ \text{E}+A_2+T_1+T_2 $ | | ||
+ | ^ $ T_2 $ | $ T_2 $ | $ T_1 $ | $ T_1+T_2 $ | $ \text{E}+A_2+T_1+T_2 $ | $ \text{E}+A_1+T_1+T_2 $ | | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Sub Groups with compatible settings ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | * [[physics_chemistry: | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Super Groups with compatible settings ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | * [[physics_chemistry: | ||
+ | |||
+ | ### | ||
+ | |||
+ | ===== Invariant Potential expanded on renormalized spherical Harmonics ===== | ||
+ | |||
+ | ### | ||
+ | |||
+ | Any potential (function) can be written as a sum over spherical harmonics. | ||
+ | $$V(r, | ||
+ | Here $A_{k, | ||
+ | The presence of symmetry induces relations between the expansion coefficients such that $V(r, | ||
+ | |||
+ | ### | ||
+ | |||
+ | ==== Expansion ==== | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | -i B(3,2) & k=3\land m=-2 \\ | ||
+ | i B(3,2) & k=3\land m=2 \\ | ||
+ | | ||
+ | | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | ==== 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_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, A(0,0)} , | ||
+ | | ||
+ | {3, 2, (I)*(B(3, | ||
+ | {4, 0, A(4,0)} , | ||
+ | | ||
+ | {4, 4, (sqrt(5/ | ||
+ | {6, 0, A(6,0)} , | ||
+ | | ||
+ | {6, 4, (-1)*((sqrt(7/ | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | ==== 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}{ -\frac{1}{7} i \sqrt{3} \text{Bpd}(3, | ||
+ | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} i \sqrt{6} \text{Bpd}(3, | ||
+ | ^$ {Y_{0}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \text{Add}(0, | ||
+ | ^$ {Y_{1}^{(2)}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} i \sqrt{6} \text{Bpd}(3, | ||
+ | ^$ {Y_{2}^{(2)}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ \frac{1}{7} i \sqrt{3} \text{Bpd}(3, | ||
+ | ^$ {Y_{-3}^{(3)}} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ -\frac{1}{3} \sqrt{\frac{10}{21}} \text{Apf}(4, | ||
+ | ^$ {Y_{-2}^{(3)}} $|$\color{darkred}{ -\frac{i \text{Bsf}(3, | ||
+ | ^$ {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}{ \frac{i \text{Bsf}(3, | ||
+ | ^$ {Y_{3}^{(3)}} $|$\color{darkred}{ 0 }$|$ -\frac{1}{3} \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_{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 }$| | ||
+ | ^$ 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{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_{\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_{\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 }$| | ||
+ | ^$ 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_{x\left(5x^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 }$|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| | ||
+ | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| | ||
+ | ^$ f_{z\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 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(y^2-z^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| | ||
+ | ^$ f_{y\left(z^2-x^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| | ||
+ | ^$ f_{z\left(x^2-y^2\right)} $|$\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 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | ==== One particle coupling on a basis of symmetry adapted functions ==== | ||
+ | |||
+ | ### | ||
+ | |||
+ | After rotation we find | ||
+ | |||
+ | ### | ||
+ | |||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ \text{s} $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ | ||
+ | ^$ \text{s} $|$ \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_{x^2-y^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ \text{Add}(0, | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$ 0 $|$ \text{Add}(0, | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3, | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3, | ||
+ | ^$ d_{\text{xy}} $|$ 0 $|$\color{darkred}{ 0 }$|$\color{darkred}{ 0 }$|$\color{darkred}{ -\frac{1}{7} \sqrt{6} \text{Bpd}(3, | ||
+ | ^$ f_{\text{xyz}} $|$\color{darkred}{ -\sqrt{\frac{2}{7}} \text{Bsf}(3, | ||
+ | ^$ f_{x\left(5x^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ \frac{4 \text{Apf}(4, | ||
+ | ^$ f_{y\left(5y^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ \frac{4 \text{Apf}(4, | ||
+ | ^$ f_{z\left(5z^2-r^2\right)} $|$\color{darkred}{ 0 }$|$ 0 $|$ 0 $|$ \frac{4 \text{Apf}(4, | ||
+ | ^$ f_{x\left(y^2-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 $|$ \text{Aff}(0, | ||
+ | ^$ f_{y\left(z^2-x^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_{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 $|$ 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_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, Ea1} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ | ||
+ | ^$ {Y_{0}^{(0)}} $|$ \text{Ea1} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ \text{s} $ ^ | ||
+ | ^$ \text{s} $|$ \text{Ea1} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Rotation matrix used** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{0}^{(0)}} $ ^ | ||
+ | ^$ \text{s} $|$ 1 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ea1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for p orbitals ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | 0 & \text{True} | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, Et2} } | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
+ | ^$ {Y_{-1}^{(1)}} $|$ \text{Et2} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ \text{Et2} $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ 0 $|$ \text{Et2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ p_x $ ^ $ p_y $ ^ $ p_z $ ^ | ||
+ | ^$ p_x $|$ \text{Et2} $|$ 0 $|$ 0 $| | ||
+ | ^$ p_y $|$ 0 $|$ \text{Et2} $|$ 0 $| | ||
+ | ^$ p_z $|$ 0 $|$ 0 $|$ \text{Et2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Rotation matrix used** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-1}^{(1)}} $ ^ $ {Y_{0}^{(1)}} $ ^ $ {Y_{1}^{(1)}} $ ^ | ||
+ | ^$ p_x $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $| | ||
+ | ^$ p_y $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $| | ||
+ | ^$ p_z $|$ 0 $|$ 1 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for d orbitals ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, (1/ | ||
+ | {4, 0, (21/10)*(Ee + (-1)*(Et2))} , | ||
+ | | ||
+ | {4, 4, (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)}} $|$ \frac{\text{Ee}+\text{Et2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ee}-\text{Et2}}{2} $| | ||
+ | ^$ {Y_{-1}^{(2)}} $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ \text{Ee} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $| | ||
+ | ^$ {Y_{2}^{(2)}} $|$ \frac{\text{Ee}-\text{Et2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ee}+\text{Et2}}{2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ | ||
+ | ^$ d_{x^2-y^2} $|$ \text{Ee} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$ \text{Ee} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $| | ||
+ | ^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Rotation matrix used** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
+ | ^$ d_{x^2-y^2} $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $| | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ -\frac{1}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ d_{\text{xy}} $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ee}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Ee}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for f orbitals ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | | ||
+ | 0 & (k\neq 4\land k\neq 6)\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ | ||
+ | | ||
+ | | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{0, 0, (1/7)*(Ea2 + (3)*(Et1 + Et2))} , | ||
+ | {4, 0, (-3/ | ||
+ | | ||
+ | {4, 4, (-3/ | ||
+ | {6, 0, (39/ | ||
+ | | ||
+ | {6, 4, (-39/ | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <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}{8} (3 \text{Et1}+5 \text{Et2}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1}) $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{-2}^{(3)}} $|$ 0 $|$ \frac{\text{Ea2}+\text{Et1}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Et1}-\text{Ea2}}{2} $|$ 0 $| | ||
+ | ^$ {Y_{-1}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} (5 \text{Et1}+3 \text{Et2}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1}) $| | ||
+ | ^$ {Y_{0}^{(3)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(3)}} $|$ \frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} (5 \text{Et1}+3 \text{Et2}) $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{2}^{(3)}} $|$ 0 $|$ \frac{\text{Et1}-\text{Ea2}}{2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{\text{Ea2}+\text{Et1}}{2} $|$ 0 $| | ||
+ | ^$ {Y_{3}^{(3)}} $|$ 0 $|$ 0 $|$ \frac{1}{8} \sqrt{15} (\text{Et2}-\text{Et1}) $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{8} (3 \text{Et1}+5 \text{Et2}) $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ | ||
+ | ^$ f_{\text{xyz}} $|$ \text{Ea2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(5x^2-r^2\right)} $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{y\left(5y^2-r^2\right)} $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et2} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(y^2-z^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1} $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{y\left(z^2-x^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1} $|$ 0 $| | ||
+ | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Et1} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <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_{\text{xyz}} $|$ 0 $|$ \frac{i}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i}{\sqrt{2}} $|$ 0 $| | ||
+ | ^$ f_{x\left(5x^2-r^2\right)} $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $| | ||
+ | ^$ f_{y\left(5y^2-r^2\right)} $|$ -\frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ -\frac{i \sqrt{5}}{4} $| | ||
+ | ^$ f_{z\left(5z^2-r^2\right)} $|$ 0 $|$ 0 $|$ 0 $|$ 1 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ f_{x\left(y^2-z^2\right)} $|$ -\frac{\sqrt{3}}{4} $|$ 0 $|$ -\frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{5}}{4} $|$ 0 $|$ \frac{\sqrt{3}}{4} $| | ||
+ | ^$ f_{y\left(z^2-x^2\right)} $|$ -\frac{i \sqrt{3}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ \frac{i \sqrt{5}}{4} $|$ 0 $|$ -\frac{i \sqrt{3}}{4} $| | ||
+ | ^$ f_{z\left(x^2-y^2\right)} $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{\sqrt{2}} $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Irriducible representations and their onsite energy** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | ^ ^$$\text{Ea2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et2}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | ^ ^$$\text{Et1}$$ | {{: | ||
+ | |$$\psi(\theta, | ||
+ | |$$\psi(\hat{x}, | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ===== Coupling between two shells ===== | ||
+ | |||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | Click on one of the subsections to expand it or < | ||
+ | |||
+ | ### | ||
+ | |||
+ | ==== Potential for s-f orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ | ||
+ | i \sqrt{\frac{7}{2}} \text{Ma1} & k=3\land m=-2 \\ | ||
+ | -i \sqrt{\frac{7}{2}} \text{Ma1} & \text{True} | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{3, 2, (-I)*((sqrt(7/ | ||
+ | | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <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 $|$ -\frac{i \text{Ma1}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i \text{Ma1}}{\sqrt{2}} $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ | ||
+ | ^$ \text{s} $|$ \text{Ma1} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for p-d orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{3, 2, (-7*I)*((1/ | ||
+ | | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of spherical Harmonics** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ {Y_{-2}^{(2)}} $ ^ $ {Y_{-1}^{(2)}} $ ^ $ {Y_{0}^{(2)}} $ ^ $ {Y_{1}^{(2)}} $ ^ $ {Y_{2}^{(2)}} $ ^ | ||
+ | ^$ {Y_{-1}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ -i \text{Mt2} $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(1)}} $|$ -\frac{i \text{Mt2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i \text{Mt2}}{\sqrt{2}} $| | ||
+ | ^$ {Y_{1}^{(1)}} $|$ 0 $|$ i \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ d_{x^2-y^2} $ ^ $ d_{3z^2-r^2} $ ^ $ d_{\text{yz}} $ ^ $ d_{\text{xz}} $ ^ $ d_{\text{xy}} $ ^ | ||
+ | ^$ p_x $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $| | ||
+ | ^$ p_y $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 0 $| | ||
+ | ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt2} $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for p-f orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & k\neq 4\lor (m\neq -4\land m\neq 0\land m\neq 4) \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{4, 0, (3/ | ||
+ | | ||
+ | {4, 4, (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{1}{2} \sqrt{\frac{3}{2}} \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt2} $| | ||
+ | ^$ {Y_{0}^{(1)}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(1)}} $|$ -\frac{1}{2} \sqrt{\frac{5}{2}} \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2} \sqrt{\frac{3}{2}} \text{Mt2} $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ | ||
+ | ^$ p_x $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ p_y $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ p_z $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | ==== Potential for d-f orbital mixing ==== | ||
+ | |||
+ | <hidden **Potential parameterized with onsite energies of irriducible representations** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | ||
+ | 0 & k\neq 3\lor (m\neq -2\land m\neq 2) \\ | ||
+ | | ||
+ | | ||
+ | \end{cases}$$ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **Input format suitable for Mathematica (Quanty.nb)** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty.nb> | ||
+ | |||
+ | Akm[k_, | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | |||
+ | ### | ||
+ | |||
+ | <code Quanty Akm_Td_xyz.Quanty> | ||
+ | |||
+ | Akm = {{3,-2, (-3/ | ||
+ | {3, 2, (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_{-2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{i \text{Mt2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{-1}^{(2)}} $|$ \frac{1}{2} i \sqrt{\frac{5}{2}} \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ \frac{1}{2} i \sqrt{\frac{3}{2}} \text{Mt2} $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{0}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ {Y_{1}^{(2)}} $|$ 0 $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{3}{2}} \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{1}{2} i \sqrt{\frac{5}{2}} \text{Mt2} $| | ||
+ | ^$ {Y_{2}^{(2)}} $|$ 0 $|$ 0 $|$ 0 $|$ -\frac{i \text{Mt2}}{\sqrt{2}} $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | |||
+ | |||
+ | ### | ||
+ | |||
+ | </ | ||
+ | <hidden **The Hamiltonian on a basis of symmetric functions** > | ||
+ | |||
+ | ### | ||
+ | |||
+ | | $ $ ^ $ f_{\text{xyz}} $ ^ $ f_{x\left(5x^2-r^2\right)} $ ^ $ f_{y\left(5y^2-r^2\right)} $ ^ $ f_{z\left(5z^2-r^2\right)} $ ^ $ f_{x\left(y^2-z^2\right)} $ ^ $ f_{y\left(z^2-x^2\right)} $ ^ $ f_{z\left(x^2-y^2\right)} $ ^ | ||
+ | ^$ d_{x^2-y^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{3z^2-r^2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{yz}} $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{xz}} $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 0 $|$ 0 $|$ 0 $|$ 0 $| | ||
+ | ^$ d_{\text{xy}} $|$ 0 $|$ 0 $|$ 0 $|$ \text{Mt2} $|$ 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 | ||
+ | |||
+ | ### |