In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full ‘sign problem’. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.
In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full ‘sign problem’. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.
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.
… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.
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.
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.
Developments are made on the quantum Monte Carlo methods towards increasing the precision and the stability of the non fixed-node projector calculations of fermions. In the first part of the developments, the wavefunction correction scheme, which was developed to increase the precision of the di usion Monte Carlo (DMC) method, is applied to non fixed-node DMC to increase the precision of such fermion calculations which do not have nodal error. The benchmark calculations indicate a significant decrease of statistical error due to the usage of the correction scheme in such non fixed-node calculations. The second part of the developments is about the modifications of the wavefunction correction scheme for having a stable non fixed-node DMC algorithm for fermions. The minus signed walkers of the non fixed-node calculations are avoided by these modifications in the developed stable algorithm. However, the accuracy of the method decreases, especially for larger systems, as a result of the discussed modifications to overcome the sign instability.