Ch Xue

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Density Functional Theory (DFT)

Density Functional Theory (DFT) is a quantum mechanical modeling method that determines the properties of a many-electron system by using the electron density instead of the complex many-body wave function.

1. Why DFT?

In the microscopic world governed by quantum mechanics, the Schrödinger Equation is the ultimate authority. However, solving it for multi-electron systems is a nightmare. The wave function Ψ(r1, r2, ..., rn) depends on 3N variables, making it computationally impossible for even medium-sized molecules.

“To solve the wave function of a complex system is to battle the ‘curse of dimensionality’.”

Density Functional Theory (DFT) changes the game. Instead of focusing on the complicated wave function, it looks at the electron density ρ(r). No matter how many electrons are in the system, the density always depends on only three spatial variables.

2. The Theoretical Pillars: Hohenberg-Kohn Theorems

The success of DFT rests on two groundbreaking theorems:

  1. Existence Theorem: The ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates.
  2. Variational Principle: There exists a functional for the energy E[ρ] such that the exact ground-state energy of the system is the global minimum value of this functional.

3. From Theory to Tool: The Kohn-Sham Equations

While the HK theorems prove that a solution exists, they don’t tell us how to find it. Kohn and Sham introduced a method to map the complex interacting system onto a fictitious “non-interacting” system, leading to the KS equations:

$$ \left[-\frac{1}{2}\nabla^2 + V_{ext}(r) + V_H(r) + V_{xc}(r)\right]\psi_i(r) = \epsilon_i\psi_i(r) $$

The term Vxc(r) (Exchange-Correlation potential) is the “heart” of DFT. It accounts for all the complex many-body effects, and its approximation is where the real science (and art) lies.

4. Real-World Applications

Today, DFT is the most widely used method in materials science and computational chemistry for:

  • Structural Prediction: Finding the most stable geometry of a crystal or molecule.
  • Energy Landscapes: Calculating reaction barriers and adsorption energies.
  • Electronic Properties: Visualizing Band Structures and Density of States (DOS).

5. Conclusion

By shifting the focus from the wave function to the electron density, DFT exemplifies the philosophy of “simplification without loss of essence.” It remains a cornerstone of modern computational research.

Published by chxue

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This is a test post to verify the commit message functionality in Hexo. The commit message should include the current date and time when the post is published, or a custom message if provided through the environment variable msg.

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A Complete Guide to Building Your Personal Website with Hexo, NexT, and GitHub Pages

Creating a personal website is a fantastic way to showcase your portfolio, write blogs, and share your knowledge. In this tutorial, we will use Hexo (a fast, simple, and powerful blog framework powered by Node.js), the elegant NexT theme, and GitHub Pages for free hosting.

Prerequisites

Before we begin, ensure you have the following installed on your computer: 1. Node.js (includes npm) 2. Git 3. A GitHub Account

Step 1: Create a GitHub Repository

First, we need a place to host your website. GitHub Pages requires a specific repository name to work out of the box.

  1. Log in to your GitHub account.
  2. Click the New button to create a new repository.
  3. Repository name: This is crucial. You MUST name it your-github-username.github.io. (Replace your-github-username with your actual exact username).
  4. Make the repository Public.
  5. Click Create repository.

Step 2: Install Hexo and Initialize Your Site

Now, let’s set up Hexo on your local machine. Open your terminal or command prompt.

  1. Install Hexo CLI globally: 
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    npm install -g hexo-cli
  2. Initialize your blog directory: Navigate to the folder where you want to keep your website files and run: 
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    hexo init my-blog
    cd my-blog
    npm install

Understanding the Directory Structure

After initialization, your my-blog folder will look like this:

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my-blog/
├── _config.yml         <-- ⚠️ Site Configuration File (Hexo Config)
├── package.json        <-- Application data and installed plugins
├── scaffolds/          <-- Templates when creating new posts
├── source/             <-- 📝 Where you write your content (Markdown files)
│   ├── _drafts/
│   └── _posts/         <-- Your blog posts go here
└── themes/             <-- 🎨 Downloaded themes go here

Step 3: Install the NexT Theme

NexT is one of the most popular, clean, and feature-rich themes for Hexo.

  1. Inside your my-blog directory, download the NexT theme using Git: 
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    git clone https://github.com/next-theme/hexo-theme-next themes/next

⚠️ The Golden Rule: Distinguishing the Two _config.yml Files

The most common source of confusion in Hexo is having two main configuration files. You must understand the difference:

  1. The Site Config (Hexo Config):
    • Location: my-blog/_config.yml (Root directory)
    • Purpose: Controls site-wide settings like the title, author, URL, language, and deployment settings.
    • Action: Open this file and change the theme variable: 
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      # my-blog/_config.yml
      theme: next
  2. The Theme Config (NexT Config):
    • Location: my-blog/themes/next/_config.yml
    • Purpose: Controls the visual appearance, menu items, sidebars, social links, and integrations (like MathJax for math equations) specifically for the NexT theme.

Step 4: Perfecting LaTeX Rendering (Inline & Display Equations)

By default, Hexo’s standard Markdown renderer (hexo-renderer-marked) conflicts with LaTeX syntax (specifically, it treats underscores _ as italics, which breaks math subscripts like Xi).

Here is the bulletproof way to fix this and render beautiful math.

1. Swap the Renderer

Uninstall the default renderer and install a math-friendly one (hexo-renderer-kramed is highly recommended for this):

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npm uninstall hexo-renderer-marked --save
npm install hexo-renderer-kramed --save

2. Enable MathJax in the NexT Config

Open your Theme Config (my-blog/themes/next/_config.yml), find the math section, and enable MathJax:

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# my-blog/themes/next/_config.yml
math:
  # Default (true) will load mathjax / katex script on demand.
  # That is it only render those page which has `mathjax: true` in Front-matter.
  every_page: false

  mathjax:
    enable: true
    # Available values: mathjax | mhchem
    tags: none

3. Using LaTeX in Your Posts

Because we set every_page: false to save loading times, you must add mathjax: true to the Front-matter (the top section) of any Markdown file where you want to write formulas.

Create a new post: 

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hexo new "My Math Post"

Open source/_posts/My-Math-Post.md and write:

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---
title: My Math Post
date: 2026-04-10 10:00:00
tags:
mathjax: true  <-- MUST ADD THIS
---

Here is an inline equation: $E = mc^2$. It renders perfectly in the flow of text.

Here is a display equation block:

$$
f(x) = \int_{-\infty}^\infty \hat{f}(\xi)\,e^{2 \pi i \xi x} \,d\xi
$$

Step 5: Deploying to GitHub Pages

You have your site and your math looks great. Let’s push it to the internet.

  1. Install the deployment plugin: 

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    npm install hexo-deployer-git --save

  2. Configure Deployment: Open your Site Config (my-blog/_config.yml), scroll to the very bottom, and modify the deploy section. Add your repository URL:

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    # my-blog/_config.yml
    deploy:
      type: git
      repo: https://github.com/your-github-username/your-github-username.github.io.git
      branch: main # or master, depending on your GitHub default branch

  3. Deploy! Run the following commands to clean old files, generate the new HTML, and push it to GitHub: 

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    hexo clean
    hexo generate
    hexo deploy
    (Note: You will be prompted to log in to your GitHub account during the deploy step if you haven’t authenticated in your terminal).

Wait a few minutes, visit https://your-github-username.github.io, and your new personal website will be live!

📌 Hexo Cheat Sheet: Common Commands

Keep these commands handy as you write and maintain your blog:

  • hexo new "Post Title" (or hexo n): Creates a new .md file in source/_posts/.
  • hexo server (or hexo s): Starts a local server at http://localhost:4000 so you can preview your site before publishing.
  • hexo clean: Deletes the public/ and db.json files. Do this if your changes aren’t showing up.
  • hexo generate (or hexo g): Builds the static HTML files from your Markdown.
  • hexo deploy (or hexo d): Pushes the generated files to GitHub.
  • hexo clean && hexo g && hexo d: The holy trinity. Cleans, generates, and deploys all in one go.
  • hexo deploy --message "Your commit message" (or hexo d -m "Your commit message"): Deploy with a custom commit message.

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Welcome to Hexo! This is your very first post. Check documentation for more info. If you get any problems when using Hexo, you can find the answer in troubleshooting or you can ask me on GitHub.

Quick Start

Create a new post

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$ hexo new "My New Post"

More info: Writing

Run server

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$ hexo server

More info: Server

Generate static files

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$ hexo generate

More info: Generating

Deploy to remote sites

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$ hexo deploy

More info: Deployment

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