Fixed-node diffusion Monte Carlo (FN-DMC) is an accurate and useful method for estimating the wave function and the energy of the quantum ground state of a many-fermion system. However, it has been shown that difficulties with the method may occur when it is applied to a degenerate excited state because the nodal surface of the degenerate trial function is generally insufficient to impose the complete symmetry properties of the trial function on the FN-DMC wave function. As a result, the tiling theorem and the symmetry-constrained variational principle may be violated by FN-DMC when the excited state is degenerate. There are two practical consequences for the study of degenerate excited states: The FN-DMC energy may lie below the energy of the lowest stationary state that transforms according to the same degenerate irreducible representation as the trial function; and the convergence of the FN-DMC energy with improvements in the trial function may not be quadratic. In this paper a diffusion Monte Carlo method for degenerate excited states is presented. It provides a direct generalization of the FN-DMC method, and when applied to the study of degenerate excited states, it has the support of the tiling theorem and the symmetry-constrained variational principle. The method is applied to the lowest degenerate state of a simple test problem in which FN-DMC has been shown to violate both the tiling theorem and the symmetry-constrained variational principle. The numerical results support the assertion that this method for degenerate excited states satisfies both the tiling theorem and the symmetry-constrained variational principle.
Fixed-node diffusion Monte Carlo (FN-DMC) is an accurate and useful method for estimating the wave function and the energy of the quantum ground state of a many-fermion system. However, it has been shown that difficulties with the method may occur when it is applied to a degenerate excited state because the nodal surface of the degenerate trial function is generally insufficient to impose the complete symmetry properties of the trial function on the FN-DMC wave function. As a result, the tiling theorem and the symmetry-constrained variational principle may be violated by FN-DMC when the excited state is degenerate. There are two practical consequences for the study of degenerate excited states: The FN-DMC energy may lie below the energy of the lowest stationary state that transforms according to the same degenerate irreducible representation as the trial function; and the convergence of the FN-DMC energy with improvements in the trial function may not be quadratic. In this paper a diffusion Monte Carlo method for degenerate excited states is presented. It provides a direct generalization of the FN-DMC method, and when applied to the study of degenerate excited states, it has the support of the tiling theorem and the symmetry-constrained variational principle. The method is applied to the lowest degenerate state of a simple test problem in which FN-DMC has been shown to violate both the tiling theorem and the symmetry-constrained variational principle. The numerical results support the assertion that this method for degenerate excited states satisfies both the tiling theorem and the symmetry-constrained variational principle.
Fixed-node diffusion Monte Carlo (DMC) is a stochastic algorithm for finding the lowest energy many-fermion wave function with the same nodal surface as a chosen trial function. It has proved itself among the most accurate methods available for calculating many-electron ground states, and is one of the few approaches that can be applied to systems large enough to act as realistic models of solids. In attempts to use fixed-node DMC for excited-state calculations, it has often been assumed that the DMC energy must be greater than or equal to the energy of the lowest exact eigenfunction with the same symmetry as the trial function. We show that this assumption is not justified unless the trial function transforms according to a one-dimensional irreducible representation of the symmetry group of the Hamiltonian. If the trial function transforms according to a multidimensional irreducible representation, corresponding to a degenerate energy level, the DMC energy may lie below the energy of the lowest eigenstate of that symmetry. Weaker variational bounds may then be obtained by choosing trial functions transforming according to one-dimensional irreducible representations of subgroups of the full symmetry group.
High Resolution Image Download MS PowerPoint Slide Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C 60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.
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 develop a formalism and present an algorithm for optimization of the trial wave function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground-state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project out a multideterminant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the noninteracting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust toward the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multideterminant expansions of the trial wave function. The method can be generalized to other wave-function forms such as pfaffians. We test the method in a model system where benchmark configuration-interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal noninteracting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77, 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.
The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.
Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.