A brief introduction to the diffusion Monte Carlo method and the fixed-node approximation

TL;DR

This paper provides a beginner-friendly introduction to the diffusion Monte Carlo method and the fixed-node approximation, explaining their theoretical basis, algorithms, and practical considerations for electronic structure calculations.

Contribution

It offers a step-by-step tutorial on DMC and fixed-node approximation, including practical examples and insights into their application to fermionic systems.

Findings

Demonstrates DMC application to harmonic oscillator and hydrogen atom

Analyzes the impact of trial wavefunction nodal surfaces on energy accuracy

Discusses extending DMC to excited states with additional considerations

Abstract

Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve, hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step towards filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schr\"odinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wavefunction nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.