Variational Monte Carlo (VMC) for Quantum mechanical systems and Atomic and Molecular Systems

This project implements Variational Monte Carlo (VMC) simulations for various quantum systems, including the Hydrogen atom, Helium atom, Hydrogen molecule (H₂), Li+, Be2+, and the Quantum Harmonic Oscillator (QHO). It provides tools for defining trial wavefunctions, computing energies, optimizing variational parameters, and visualizing results.

Features

• VMC Implementations:

  • Hydrogen atom (ground state)
  • Helium atom (ground state)
  • Li+ and Be2+ ion ground state calculations
  • Hydrogen molecule (H₂) with bonding and anti-bonding wavefunctions
  • Quantum Harmonic Oscillator (QHO)

• Trial Wavefunctions: User-defined trial wavefunctions with adjustable variational parameters.

• Analytical Solutions: Exact analytical solutions for the Hydrogen atom's ground state and the QHO wavefunction for comparison.

• Jastrow Factor: Includes a Jastrow correlation factor for the H₂ trial wavefunction to account for electron-electron repulsion.

• Numerical Optimization: Optimizes variational parameters (e.g., alpha, beta) to minimize energy.

• Local Energy Calculation: Implements a finite difference method for calculating the local energy.

• Metropolis Sampling: Efficient Monte Carlo sampling using the Metropolis algorithm.

• Data Analysis & Visualization: Generates interactive plots using Plotly for better result interpretation.

Installation

Ensure the required libraries are installed:

pip install matplotlib numpy tqdm numba plotly

Clone the repository:

git clone https://github.com/SuvamT0071/QuantumMonte.git
cd VMC

Code Structure

The project is organized into separate files for different systems and functionalities:

Single-Atom Systems (e.g., Hydrogen, Helium, QHO)

• Trial Wavefunctions: Define the ground state wavefunctions (e.g., Hyd_GS, Helium_GS, QHO_GS, lightatoms_GS).
• Probability Density: Calculate the wavefunction probability density (e.g., Hyd_GSPDF, He_GSPDF, QHO_GSPDF, lightatoms_GSPDF).
• Local Energy: Compute the local energy (e.g., Hyd_local, He_loc_en, QHO_local, light_loc_en).
• VMC Simulations: Perform VMC sweeps to sample configurations and estimate energies (e.g., Hyd_VMC, Helium_VMC,lightatoms_VMC).
• Optimization: Optimize variational parameters (e.g., Hyd_alpha_opt, Helium_alpha_opt, lightatoms_alpha_opt, lightatoms_beta_opt).

Molecular Systems (e.g., H₂)

• Proton Positioning: Define proton positions for H₂ (proton_points).
• Single-Electron Wavefunction: Calculate single-electron wavefunctions (single_electron_wavefunction).
• Jastrow Factor: Implement the Jastrow correlation factor (calc_Jastrow).
• Total Wavefunction: Construct the full molecular wavefunction (total_wavefunction).
• VMC Simulations: Run VMC for H₂, including Jastrow factors and bonding/anti-bonding states (H2_VMC).
• Parameter Optimization: Optimize alpha and beta parameters for minimal energy (alpha_opt, beta_opt).

Usage

To run the notebooks, open them in Jupyter:

jupyter notebook QMC_HYD.ipynb
jupyter notebook QMC_H2_matplotlib.ipynb

Key Considerations for H₂

• Bonding vs. Anti-Bonding: Use the sign parameter in functions like single_electron_wavefunction to select bonding (sign=1) or anti-bonding (sign=-1) states.
• Optimization Order: Alpha is typically optimized before beta, but simultaneous optimization can yield better results.

Recommendations for Improvement

• Simultaneous Alpha and Beta Optimization: Use scipy.optimize.minimize for joint parameter optimization.
• Advanced Wavefunctions: Explore more complex functional forms for improved accuracy.
• Multiple Starting Points: Run optimizations from different starting points to avoid local minima.
• Increased Sample Size: Use more VMC steps for reduced statistical noise.

License

This project is licensed under the MIT License. Feel free to use, modify, and distribute the code as you see fit.

Credits:

This project was made by Suvam Tripathy, MSc Physics, IIT Madras as a part of a mini-project.

For any questions or contributions, feel free to reach out or submit a pull request. Happy coding!