In this paper, a diffusion Monte Carlo (DMC) method to determine the ground state of a particle in a double well potential is presented. The DMC method uses displacements, creations and deletions of diffusers or walkers. The effects of double well parameters on the ground state are studied. For validation of our DMC method, a harmonic oscillator potential is used and the DMC method gives accurate results. In the double well potential, the DMC results has shown that the ground state energy increases as the depth of the double well increases.
In this paper, a diffusion Monte Carlo (DMC) method to determine the ground state of a particle in a double well potential is presented. The DMC method uses displacements, creations and deletions of diffusers or walkers. The effects of double well parameters on the ground state are studied. For validation of our DMC method, a harmonic oscillator potential is used and the DMC method gives accurate results. In the double well potential, the DMC results has shown that the ground state energy increases as the depth of the double well increases.
We propose a new method to calculate ground state position dependent observables in quantum many-body systems. The method, which we call the path-integral diffusion Monte Carlo (PI-DMC) method, is essentially a combination of path-integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. The distribution resulting from a DMC simulation is further propagated in imaginary time by PIMC sampling. Tests of the new method for simple cases such as the harmonic oscillator, a double well, and a set of ten coupled harmonic oscillators show that the results for several observables converge rapidly to the exact result.
The diffusion Monte Carlo method is applied to study the ground-state properties of charged bosons in one dimension confined in a harmonic double-well trap. The particles interact repulsively through a Coulombic 1/ r potential. Numerical results show that the well separation has significant influence on the ground-state properties of the system. When the interaction of the system is weak, ground-state energy decreases with the increasing well separation and has a minimal value. If the well separation increases continually, the ground-state energy increases and approaches to a constant gradually. This effect will be abatable in the strong interacting system. In addition, by calculating the density of the systems for different interaction strengths with various well separations, we find that the density increases abnormally when the well separation is large at the centre of the system.
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.
Diffusion Monte Carlo (DMC) is a technique for obtaining the ground-state solution to the vibrational time-independent Schrödinger equation based on a stochastic sampling of an electronic potential energy surface (PES). Ideally, the electronic energies used in DMC are calculated at the CCSD(T) level of theory. Recently, Tom Miller’s group at Caltech has developed a technique for calculating accurate CCSD(T) corrections to the electronic energy at the cost of a standard Hartree-Fock calculation using Gaussian process regression, a machine learning technique\footnote{Welborn, M. , Cheng, L., Miller, T. M. JCTC. 2018 14 (9), 4772-4779}\footnote{Cheng, L., Kovachki, N. B., Welborn, M., Miller, T. M. JCTC. 2019 15 (12), 6668-6677}. DMC is a well-suited technique to validate these PESs, referred to as MOB-ML surfaces, since DMC samples the entire region of the PES in which the vibrational ground state wave function has amplitude. Despite the speed of these MOB-ML PESs relative to an \emph{ab initio} calculation, calculating the Hartree-Fock energy for every one of thousands of DMC configurations over tens of thousands of time-steps makes the use of these surfaces computationally challenging. To this end, we combined two techniques to bring these calculations into computational feasibility. First, we implemented a DMC algorithm to take advantage of massively parallel high performance computing environments. Second, we use this DMC code to collect training data for a neural network (NN). We use a generic algorithm to train the NN to learn the MOB-ML surface based on the points sampled by the DMC. Using TensorFlow, the evaluation of the NN can then easily be done using graphics processing units (GPUs). We have performed small-scale DMC calculations on \chem{H2O} and \chem{CH5+} using these MOB-ML surfaces, and have also run large-scale NN+MOB-ML DMC simulations to obtain a high resolution ground state vibrational wave function and measure of the zero-point energy. Due to the generality of both the MOB-ML surface generation and the NN fitting workflow, we can easily extend this work to larger systems, such as large ion-water clusters or a Criegee intermediate, where running the DMC with traditional electronic structure would be intractable.
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 size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.