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中国化学快报：英文版

A Novel Exact Fixed-node Quantum Monte Carlo Algorithm

HongXinHUANG

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

In this paper we proposed a novel exact fixed-node quantum Monte Carlo (EFNQMC)algorithm,which is a self-optimizing and self-improving procedure,In contrast to the previous EFNQMC method,the trial function is optimized synchronistically in the diffusion procedure,but not before the beginning of EFNQMC computation.In order to optimize the trial function,the improved steepest descent technique is used,in which the step size is automatically adjustable.The procedure is quasi-Newton and converges super linearly.We also use a novel trial function,which has correct electron-electron and electron-nucleus cusp conditions.The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1^1A1 state of CH2,^1Ag state of C8 and the ground-states of H2,LiH,Li2,H2O,respectively.The test results show that both the novel algorithm and the trial function proposed in the present paper are very excellent.

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Exact fixed‐node quantum monte carlo: Self‐optimizing procedure

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Abstract In this paper, a novel exact fixed‐node quantum Monte Carlo (EFNQMC) algorithm was proposed, which is a self‐optimizing and self‐improving procedure. In contrast to the previous EFNQMC method, the importance function of this method is optimized synchronistically in the diffusion procedure, but not before beginning the EFNQMC computation. In order to optimize the importance function, the improved steepest descent technique is used, in which the step size is automatically adjustable. The procedure is quasi‐Newton type and converges super linearly. The present method also uses a novel trial function, which has correct electron‐electron and electron‐nucleus cusp conditions. The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1 1A1 state of CH2, 1A1 state of C8 and the ground‐states of H2, LiH, Li2 and H2O.

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In this paper, a novel exact fixed-node quantum Monte Carlo (EFNQMC) algorithm was proposed, which is a self-optimizing and self-improving procedure. In contrast to the previous EFN-QMC method, the importance function of this method is optimized synchronistically in the diffusion procedure, but not be-fore beginning the EFNQMC computation. In order to optimize the importance function, the improved steepest descent tech-nique is used, in which the step size is automatically adjustable.The procedure is quasi-Newton type and converges super linear-ly. The present method also uses a novel trial function, which has correct electron-electron and electron-nucleus cusp condi-tious. The novel EFNQMC algorithm and the novel trial func-tion are employed to calculate the energies of 1 ^1A1 state of CH2, ^1Ag state of Cs and the ground-states of H2, LiH, Li2 and H2O.

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In this paper we proposed a novel exact fixed-node quantum Monte Carlo (EFNQMC)algorithm,which is a self-optimizing and self-improving procedure,In contrast to the previous EFNQMC method,the trial function is optimized synchronistically in the diffusion procedure,but not before the beginning of EFNQMC computation.In order to optimize the trial function,the improved steepest descent technique is used,in which the step size is automatically adjustable.The procedure is quasi-Newton and converges super linearly.We also use a novel trial function,which has correct electron-electron and electron-nucleus cusp conditions.The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1^1A1 state of CH2,^1Ag state of C8 and the ground-states of H2,LiH,Li2,H2O,respectively.The test results show that both the novel algorithm and the trial function proposed in the present paper are very excellent.
Abstract In this paper, a novel exact fixed‐node quantum Monte Carlo (EFNQMC) algorithm was proposed, which is a self‐optimizing and self‐improving procedure. In contrast to the previous EFNQMC method, the importance function of this method is optimized synchronistically in the diffusion procedure, but not before beginning the EFNQMC computation. In order to optimize the importance function, the improved steepest descent technique is used, in which the step size is automatically adjustable. The procedure is quasi‐Newton type and converges super linearly. The present method also uses a novel trial function, which has correct electron‐electron and electron‐nucleus cusp conditions. The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1 1A1 state of CH2, 1A1 state of C8 and the ground‐states of H2, LiH, Li2 and H2O.
In this paper, a novel exact fixed-node quantum Monte Carlo (EFNQMC) algorithm was proposed, which is a self-optimizing and self-improving procedure. In contrast to the previous EFN-QMC method, the importance function of this method is optimized synchronistically in the diffusion procedure, but not be-fore beginning the EFNQMC computation. In order to optimize the importance function, the improved steepest descent tech-nique is used, in which the step size is automatically adjustable.The procedure is quasi-Newton type and converges super linear-ly. The present method also uses a novel trial function, which has correct electron-electron and electron-nucleus cusp condi-tious. The novel EFNQMC algorithm and the novel trial func-tion are employed to calculate the energies of 1 ^1A1 state of CH2, ^1Ag state of Cs and the ground-states of H2, LiH, Li2 and H2O.
We propose an algorithm for optimizing quantum Monte Carlo wave functions. An improved steepest-descent technique is used, with step size automatically adjustable to obtain a procedure that converges superlinearly. We also propose a novel trial function, which has both correct electron–electron cusp conditions and correct electron–nucleus cusp conditions. To test the optimization procedure and the optimized trial function, the ground states for CH4 and H2O molecules were investigated using variational Monte Carlo (VMC) and fixed-node quantum Monte Carlo (FNQMC) calculations. For CH4 and H2O, the VMC recovered 73.3% and 57.9% of the correlation energy, respectively, and the FNQMC recovered 99.3% and 92.8%, respectively. The optimization procedure is three to five times faster than conventional usual steepest-descent procedures. The trial functions optimized are more accurate than prior trial functions of similar complexity.
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.
A reformulation of the fixed-node diffusion quantum Monte Carlo method (FN-DQMC) in terms of the N-particle density matrix is presented, which allows us to reduce the computational effort to linear for the evaluation of the local energy. The reformulation is based on our recently introduced density matrix-based approach for a linear-scaling variational QMC method [J. Kussmann et al., Phys. Rev. B. 75, 165107 (2007)]. However, within the latter approach of using the positive semi-definite N-particle trial density (rhoN T(R)=mid R:Psi(T)(R)mid R:(2)), the nodal information of the trial function is lost. Therefore, a straightforward application to the FN-DQMC method is not possible, in which the sign of the trial function is usually traced in order to confine the random walkers to their nodal pockets. As a solution, we reformulate the FN-DQMC approach in terms of off-diagonal elements of the N-particle density matrix rhoN T(R;R'), so that the nodal information of the trial density matrix is obtained. Besides all-electron moves, a scheme to perform single-electron moves within N-PDM QMC is described in detail. The efficiency of our method is illustrated for exemplary calculations.
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.