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Molecular Physics

Quantum simulation of systems with nodal surfaces

D. F. Coker, R.O. Watts

Published Aug 20, 1986• 66 citations
Publisher

Abstract

A modification of the diffusion Monte Carlo algorithm that enables the direct simulation of wave functions containing nodal surfaces is described. The use of orthogonality constraints during the simulation enables the properties of excited state systems to be calculated. For Fermi systems, spin variables and exchange operators are also included in the algorithm. The effectiveness of the methods is demonstrated using applications to several simple models.

DOI
10.1080/00268978600101841

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A modification of the diffusion Monte Carlo algorithm that enables the direct simulation of wave functions containing nodal surfaces is described. The use of orthogonality constraints during the simulation enables the properties of excited state systems to be calculated. For Fermi systems, spin variables and exchange operators are also included in the algorithm. The effectiveness of the methods is demonstrated using applications to several simple models.
This article describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calculate the properties of many-electron systems. These stochastic wave-function-based approaches provide a very direct treatment of quantum many-body effects and serve as benchmarks against which other techniques may be compared. They complement the less demanding density-functional approach by providing more accurate results and a deeper understanding of the physics of electronic correlation in real materials. The algorithms are intrinsically parallel, and currently available high-performance computers allow applications to systems containing a thousand or more electrons. With these tools one can study complicated problems such as the properties of surfaces and defects, while including electron correlation effects with high precision. The authors provide a pedagogical overview of the techniques and describe a selection of applications to ground and excited states of solids and clusters.
Treating the fermionic ground state-problem as a constrained stochastic optimization problem, a formalism for fermionic quantum Monte Carlo is developed that makes no reference to a trial wave function. Exchange symmetry is enforced by nonlocal terms appearing in the Green's function corresponding to an additional walker propagation channel. Complemented by a treatment of diffusion that encourages the formation of a stochastic nodal surface, we find that an approximate long-range extension of walker cancellations can be employed without introducing significant bias, reducing the number of walkers required for a stable calculation. A proof-of-concept implementation is shown to give a stable fermionic ground state for simple harmonic and atomic systems.
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
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 toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger 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 wave function 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.
We study several aspects of the recently introduced fixed-phase spinor diffusion Monte Carlo method, in particular, its relation to the fixed-node method and its potential use as a general approach for electronic structure calculations. We illustrate constructions of spinor-based wave functions with the full space-spin symmetry without assigning up or down spin labels to particular electrons, effectively "complexifying" even ordinary real-valued wave functions for Hamiltonians without spin terms. Interestingly, with proper choice of the simulation parameters and spin variables, such fixed-phase calculations enable one to reach also the fixed-node limit. The fixed-phase approximation has several desirable properties when compared to the fixed-node approximation. The fixed-phase solution provides a straightforward interpretation as the lowest bosonic state in a given effective potential generated by the many-body approximate phase, whereas nodal boundary conditions are defined through less intuitive and complicated hypersurfaces with one dimension less than the original configuration space. In addition, the divergences of the local energy and drift at real wave function nodes are smoothed out to lower dimensionality when the wave function is complexified, thus decreasing the variation of sampled quantities and eliminating artificial nodal domain issues that can occur in the fixed-node formalism. We illustrate some of these properties on calculations of selected first-row systems that recover the fixed-node results with quantitatively similar levels of the corresponding biases. At the same time, the fixed-phase approach opens new possibilities for more general trial wave functions with further opportunities for increasing accuracy in practical calculations.