update

Author: Harrison-Li

docs/Atomic simulation of materials/DFT in solids.md

@@ -4,7 +4,7 @@
 
 **1. Kohn-sham equation (Auxiliary method)**
 $$
-(-\frac{1}{2}\grad + \underset{V_{eff}(\vec{r_1})}{\underbrace{{V_{ne} +V_H+V_{xc}}}})\, \theta_i (\vec{r_1}) = \epsilon_i \theta_i(\vec{r_i})
+(-\frac{1}{2}\nabla + \underset{V_{eff}(\vec{r_1})}{\underbrace{{V_{ne} +V_H+V_{xc}}}})\, \theta_i (\vec{r_1}) = \epsilon_i \theta_i(\vec{r_i})
 $$
 **2. SCF Loop**
 
@@ -20,9 +20,9 @@ e.g. VWN, PZ, PWQ, SPZ
 
 Effective but over-binding
 
-- **GGA:** add $\grad \rho(\vec{r})$ (Non-homogeneity)
+- **GGA:** add $\nabla \rho(\vec{r})$ (Non-homogeneity)
 
-$E_\text{xc}^\text{GGA} = \int f(\rho,\grad \rho)d\vec{r}= E_\text{x}^\text{GGA}(\rho) +E_\text{c}^\text{GGA}(\rho)$
+$E_\text{xc}^\text{GGA} = \int f(\rho,\nabla \rho)d\vec{r}= E_\text{x}^\text{GGA}(\rho) +E_\text{c}^\text{GGA}(\rho)$
 
 e.g. PBE, BLYP, ...
 

docs/Atomic simulation of materials/DFT.md

@@ -64,7 +64,7 @@ Previous section describes how to represent multi-electron system into density f
 
 The actual density functional theory calculations are performed on the auxiliary single-electron independent-particle system defined by the mono-electronic auxiliary hamiltonian $\hat{H}_\text{aux}$, also known as the Kohn-Sham operator $\hat{F}_\text{KS}$ in $a.u.:^2$
 $$
-\hat{H}_\text{aux}=\hat{F}_\text{KS}=-\frac{1}{2}\grad^2+V_\text{eff}(r)
+\hat{H}_\text{aux}=\hat{F}_\text{KS}=-\frac{1}{2}\nabla^2+V_\text{eff}(r)
 $$
 
 \[

docs/Atomic simulation of materials/DFT2.md

@@ -15,7 +15,7 @@
 
 **Kohn-Sham auxiliary system:**
 
-$\hat{H}_\text{aux}=\hat{F}_\text{KS}=-\frac{1}{2}\grad^2+V_\text{eff}(\vec{r})$
+$\hat{H}_\text{aux}=\hat{F}_\text{KS}=-\frac{1}{2}\nabla^2+V_\text{eff}(\vec{r})$
 
 ------
 
@@ -39,7 +39,7 @@ Again, the unknown term is $E_{\mathrm{xc}}[\rho]$. Similarly as what we done in
 Recall that, variational principle $\theta_i(r)\rightarrow \delta \theta_i(r), \delta \text{ is the variation factor}$ 
 $$
 \hat{F}^\text{KS}\theta_i(r_1)=\epsilon \theta_i (r_1)\\
--\frac{1}{2}\grad_1^2 +
+-\frac{1}{2}\nabla_1^2 + V_\text{eff}(r_1)
 $$
 
 
@@ -87,6 +87,22 @@ $$
 $$
 **Generalized Gradient Approximation (GGA)**
 $$
-E_\text{xc}^\text{GGA}=\int f(\rho, \grad\rho)dr
+E_\text{xc}^\text{GGA}=\int f(\rho, \nabla\rho)dr
 $$
-The gradient itself It's more like adding a Tylor expansion, the LDA method is the zero order of the series, while the gradient behaves as polynomials, making it more close the to value.
+The gradient itself It's more like adding a Tylor expansion, the LDA method is the zero order of the series, while the gradient behaves as polynomials, making it more close the to value.
+
+**Meta-GGA**
+Rung-3 functional on Jacob’s ladder of DFT. It improves over GGA by using additional **semilocal** information about the electron density:
+$$
+E_\text{xc}^\text{meta-GGA} = \int f(\rho,\nabla \rho, \nabla^2 \rho,\tau)
+$$
+
+- $\tau = \frac{1}{2}\sum_i|\nabla\psi_i|^2 \rightarrow$ kinetic-energy density
+
+Kinetic energy density physical meanings:
+
+It is large where orbitals oscillate rapidly (e.g. core, bonding), small where density is smooth.
+
+1. **bonding type** (metallic, ionic, covalent)
+2. **iso-orbital regions** (single-orbital like H atom)
+3. **weak interactions** (vdW regions)

docs/Atomic simulation of materials/Hartree-Fock.md

@@ -13,7 +13,7 @@ For two electron integral:  $\epsilon_i=\int\mathcal{X_1}^*(x_1)\mathcal{X_2}^*(
 **Energy contribution**
 
 - Coulomb: $J =\int\mathcal{X_1}^*(x_1)\mathcal{X_2}^*(x_2)\frac{1}{|r_{12}|}\mathcal{X_1}(x_1)\mathcal{X_2}(x_2)dx_1dx_2 = \int|\mathcal{X_1}(x_1)^2|\frac{1}{r_{12}}|\mathcal{X_2}(x_2)^2|dx_1dx_2$
-- Exchange: $K =$
+- Exchange: $K =\int\mathcal{X_1}^*(x_1)\mathcal{X_2}^*(x_2)\frac{1}{|r_{12}|}\mathcal{X_1}(x_2)\mathcal{X_2}(x_1)dx_1dx_2 $
 
 
 

docs/ML For Materials/JT-VAE_handwritten_derivation_view.md

@@ -7,16 +7,16 @@ This document contains handwritten mathematical derivations for the Junction Tre
 You can view or download the full PDF document:
 
 <div style="text-align: center; margin: 20px 0;">
-  <a href="JT-VAE_handwritten_derivation.pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
+  <a href="docs/ML For Materials/JT-VAE_handwritten_derivation.pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
     📄 Open Handwritten Derivation PDF
   </a>
 </div>
 
-## Embedded View
+## PDF Preview
 
-<iframe src="JT-VAE_handwritten_derivation.pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
+<iframe src="docs/ML For Materials/JT-VAE_handwritten_derivation.pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
   <p>Your browser does not support PDFs. 
-  <a href="JT-VAE_handwritten_derivation.pdf">Download the PDF</a> instead.</p>
+  <a href="docs/ML For Materials/JT-VAE_handwritten_derivation.pdf">Download the PDF</a> instead.</p>
 </iframe>
 
 ---

docs/ML/VAE.md

@@ -7,16 +7,16 @@ This document contains detailed information about Variational Autoencoders.
 You can view or download the full PDF document:
 
 <div style="text-align: center; margin: 20px 0;">
-  <a href="VAE .pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
+  <a href="docs/ML/VAE .pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
     📄 Open VAE.pdf
   </a>
 </div>
 
-## Embedded View
+## PDF Preview
 
-<iframe src="VAE .pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
+<iframe src="docs/ML/VAE .pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
   <p>Your browser does not support PDFs. 
-  <a href="VAE .pdf">Download the PDF</a> instead.</p>
+  <a href="docs/ML/VAE .pdf">Download the PDF</a> instead.</p>
 </iframe>
 
 ---

docs/Python/creating-reading-and-writing_view.md

@@ -7,16 +7,16 @@ This PDF tutorial covers creating, reading, and writing data with Pandas.
 You can view or download the full PDF document:
 
 <div style="text-align: center; margin: 20px 0;">
-  <a href="creating-reading-and-writing.pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
+  <a href="docs/Python/creating-reading-and-writing.pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
     📄 Open Creating/Reading/Writing Tutorial PDF
   </a>
 </div>
 
-## Embedded View
+## PDF Preview
 
-<iframe src="creating-reading-and-writing.pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
+<iframe src="docs/Python/creating-reading-and-writing.pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
   <p>Your browser does not support PDFs. 
-  <a href="creating-reading-and-writing.pdf">Download the PDF</a> instead.</p>
+  <a href="docs/Python/creating-reading-and-writing.pdf">Download the PDF</a> instead.</p>
 </iframe>
 
 ---

docs/Python/indexing-selecting-assigning_view.md

@@ -7,16 +7,16 @@ This PDF tutorial covers indexing, selecting, and assigning operations in Pandas
 You can view or download the full PDF document:
 
 <div style="text-align: center; margin: 20px 0;">
-  <a href="indexing-selecting-assigning.pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
+  <a href="docs/Python/indexing-selecting-assigning.pdf" target="_blank" style="background-color: #4CAF50; color: white; padding: 15px 32px; text-align: center; text-decoration: none; display: inline-block; font-size: 16px; margin: 4px 2px; cursor: pointer; border-radius: 4px;">
     📄 Open Indexing Tutorial PDF
   </a>
 </div>
 
-## Embedded View
+## PDF Preview
 
-<iframe src="indexing-selecting-assigning.pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
+<iframe src="docs/Python/indexing-selecting-assigning.pdf" width="100%" height="800px" style="border: 1px solid #ccc;">
   <p>Your browser does not support PDFs. 
-  <a href="indexing-selecting-assigning.pdf">Download the PDF</a> instead.</p>
+  <a href="docs/Python/indexing-selecting-assigning.pdf">Download the PDF</a> instead.</p>
 </iframe>
 
 ---