This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time-averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many-fermion systems, where the Pauli principle must be taken into account, we can use the fixed-node approximation, but we also discuss the potentially exact release-node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), associated with a possibly refined choice of the reference function.
This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time-averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many-fermion systems, where the Pauli principle must be taken into account, we can use the fixed-node approximation, but we also discuss the potentially exact release-node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), associated with a possibly refined choice of the reference function.
Abstract These two papers describe the development of a variant of diffusion QMC with importance sampling and its applications to a few simple systems. The method is based on the generalized FeynmanKac path integral formalisma and is termed the “full generalized Feynman-Kac” or FGFK method. Its major advantage, relative to other diffusion-based methods, is the elimination of branching or multiplication of walkers. Its major disadvantage is a greater complexity. The basic implementation is very nearly the same as that of standard diffusion QMC with importance sampling, and it incorporates diffusion with drift in the same way. The uses of trial functions, fixed nodes, extrapolation to zero time-step size, and other procedures except for branching are identical. However, the evaluation of mean quantities can be carried out in terms of ‘lj;2 sampling to obtain energies and other properties based on the FGFK formula. The applications described include ground and excited states of the harmonic oscillator and the helium atom and the ground state of the hydrogen molecule. For these the energies as well as a dozen other observables such as the averages of interelectron distances were determined. The results were found to be in good agreement with accepted values. A combination of FGFK with the released-node method was investigated for the case of the one-dimensional harmonic oscillator and found to be successful for that case.
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.
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.
We present a numerically exact procedure for the calculation of an important class of finite temperature quantum mechanical time correlation functions. The present approach is based around the stationary phase Monte Carlo (SPMC) method, a general mathematical tool for the calculation of high dimensional averages of oscillatory integrands. In the present context the method makes possible the direct numerical path integral calculation of real-time quantum dynamical quantities for times appreciably greater than the thermal time (βℏ). Illustrative applications involving finite temperature anharmonic motion are presented. Issues of importance with respect to future applications are identified and discussed.
It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.
For distinguishable particles it is well known that Brownian motion and a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In order to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a noninteracting system. For motion in one dimension, we emphasize that the permutation symmetry of the identical particles completely determines the domain of Brownian motion and the appropriate boundary conditions: absorption for fermions, reflection for bosons. Further analysis of the sample paths for motion in three dimensions allows us to decompose these paths into a superposition of one-dimensional sample paths. This reduction expresses the propagator (and consequently the energy and other thermodynamical quantities) in terms of well-behaved one-dimensional fermion and boson diffusion processes and the Feynman-Kac functional.