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.
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.
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.
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.
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.
The diffusion quantum Monte Carlo method (DMC) provides a numeric solution of the time dependent Schrödinger equation by means of a random walk process. To fulfil the pauli principle the nodes of a so called trial wavefunction, obtained with standard ab initio quantum chemistry methods like HF, DFT, or CASSCF, are considered as additional boundary conditions. In case of such nodes coincide with those of the exact wavefunction DMC yields the exact solution of the Schrödinger equation. Generally, this is not true and one ends up with an systematic bias, referred to as the node location error. Theoretical chemistry aims the prediction of reaction, excitation, or activation energies, i.e. energy differences. Therefore, the node location error has to be seen under the light of efficient error cancellation. One can distinguish dynamic and non-dynamic electron correlation. The first one accounts for the instantaneous, short distance interaction among the electrons. The non-dynamic correlation addresses the interaction between different states of the same symmetry. Both kinds of correlation have a significant impact on the nodal structure and therefore govern the node location error. In this thesis the isomerization of bicylo[1.1.0]butane has been studied by exploiting the DMC method. The underlying reaction mechanism poses a great challenge to quantum chemistry because it comprises species with different amounts of non-dynamical correlation. Indeed, very sophisticated methods like for example CCSD(T) fail in giving a proper description. For the DMC method, the construction of the trial function is of crucial importance. Hence, the latter is generated by means of CASSCF calculations. With regard to efficient error cancellation it is most reasonable to take only those CSFs into account which contribute significantly to the proper description of the non-dynamic correlation. Since the DMC method is variational for the fermionic ground state, the incorporation of such a CSFs leads to a decrease of the energy compared with the output of a computation just based on the reference determinant. In the present case, only up to three CSFs very necessary. As a consequence, the direct optimization of the DMC energy with respect to the CASSCF coefficients is possible. By means of this construction scheme, dynamic and non-dynamic electron correlation effects are accounted for in equal measure for every molecule comprising the reaction path. Hence, the error cancellation is excellent. The obtained reaction, and activation energies are in very good agreement with experimental data and completely renormalized coupled cluster calculations. The presented CSF selection scheme yields very short linear combinations of CSFs and therefore should be applicable even for larger organic molecules. As it has already pointed out, quantum chemistry aims the prediction of energy differences rather than absolute values. A very efficient approach to calculate energy differences within a Monte Carlo framework is correlated sampling. The main idea behind this procedure is to describe the two different systems whose energy difference is to be calculated by means of the same random walk process. The resulting statistical correlation gives rise to a reduction of the statistical uncertainty of the energy difference. As far as we know, until now no implementation of the correlated sampling formalism into the DMC method has been reported in the literature. To implement correlated sampling in the DMC method a mathematical formalism based on the theory of stochastic differential equations has been devised. Based on this formalism an appropriate algorithm has been implemented for test systems like the particle in a one dimensional box, and the harmonic oszillator. Correlated sampling is very well suited for the computation of small energy differences with considerable small statistical uncertainties. A possible application would be the calculation of the weak interaction between DNA base pairs. For this purpose the corresponding formalism has to be implemented in our quantum Monte Carlo program Amolqc. In doing so, a major task is the implementation of routines which account for the propagation with respect to a positive guide function what has already been finished in the course of this work.
… 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.