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Numerical Chapter
Published in James J Y Hsu, Nanocomputing, 2017
Well explained in the lecture notes by James B.Anderson (2002) are the Diffusion Quantum Monte Carlo (DQMC) and Green’s Function Quantum Monte Carlo (GFQMC) Methods. The DQMC method is based on the similarity of the Schrodinger equation and the diffusion equation: ∂Ψ(r⇀,τ)∂τ=12∇2Ψ(r⇀,τ)-VΨ(r⇀,τ), $$ \frac{{\partial \Psi (\mathop {\overset{{\rightharpoonup}} {r}}\limits^{{}} ,\tau )}}{{\partial \tau}} = \frac{1}{2}\nabla ^{2} \Psi (\mathop {\overset{{\rightharpoonup}} {r}}\limits^{{}} ,\tau ) - V\Psi (\mathop {\overset{{\rightharpoonup}} {r}}\limits^{{}} ,\tau ), $$
Introduction
Published in Paolo Di Sia, Mathematics and Physics for Nanotechnology, 2019
Monte Carlo methods are a class of stochastic optimisation techniques that are particularly reliable for the transition probabilities of statistical mechanics and quantum physics; they are therefore applied to many problems, not only to the commonly considered optimisation problems. The quantum Monte Carlo method is one of the most accurate methods for the electronic structure, extendable to systems in the nanorange. It numerically solves with great accuracy the Schrödinger equation. Kinetic Monte Carlo methods are an important class of techniques that deal with a range of atomic particles and aggregate phenomena, in addition to those of the electronic structure. Also the computational geometry is a mature area of computer science algorithmics.
Introduction to Computational Methods in Nanotechnology
Published in Sarhan M. Musa, ®, 2018
Studies of the above-mentioned two nano-building blocks, molecular magnets and semiconductor quantum dots, offer great promise for the future but will require new theoretical approaches and computationally intensive studies. Current algorithms and numerical methods must be made more efficient and new ones should be invented. Currently, molecular dynamics simulation methods can handle systems with tens of thousands of atoms; however, to fully exploit their power, algorithms need to be made scalable and fully parallelized Computational methods are especially useful in providing benchmarks for nano-building blocks, where experimental data are unreliable or hard to reproduce. Lack of clear prescriptions for obtaining reliable results in nano blocks is another challenging problem for experiment and theory. While new experiments will need to be designed to ensure reproducibility and the validity of the measurements, the theoretical challenge is to construct new theories that would cross-check such conclusions. While great strides have been made in various simulation methods, a number of fundamental issues remain. In particular, the diversity of time and length scales remains a great challenge at the nanoscale. Intrinsic quantum attributes like transport and charge transfer remain a challenge to incorporate into classical description. For example, at best, quantum Monte Carlo (QMC) simulation methods are effective and easy to implement only at zero or very low temperatures. Even though the QMC method and its many variants are currently the most accurate methods that can be extended to systems in the nanoscience range, major improvements are needed.
The paradigm of complex probability and Monte Carlo methods
Published in Systems Science & Control Engineering, 2019
Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman-Kac path integrals (Assaraf, Caffarel, & Khelif, 2000; Caffarel, Ceperley, & Kalos, 1993; Del Moral, 2003; Del Moral, 2004; Del Moral & Miclo, 2000a, 2000b; Hetherington, 1984). The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean field particle interpretation of neutron-chain reactions (Fermi & Richtmyer, 1948), but the first heuristic-like and genetic type particle algorithm (also known as Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984 (Hetherington, 1984) In molecular chemistry, the use of genetic heuristic-like particle methodologies (also known as pruning and enrichment strategies) can be traced back to 1955 with the seminal work of Rosenbluth and Rosenbluth (1955).
Equation of State: Manhattan Project Developments and Beyond
Published in Nuclear Technology, 2021
Scott D. Crockett, Franz J. Freibert
A modern EOS model is built by leveraging data collected over experimental conditions ranging from ambient, static compression, and shock regimes, and by developing an integrated theoretical approach to ensure data set inclusion. These modern models cover compressions of 0 to 106 volumetric strains and temperatures from 0 to 109 kelvins. Once optimized across multiple data sets, the model forms naturally extend to known thermodynamic limits. We start, however, with ambient data along the 1-atm isobar. The EOS-relevant thermophysical properties data include information for the reference density, thermal expansion, specific heat, and bulk moduli. X-ray diffraction measures the initial crystal structure and density. Dilatometry measures the thermal expansion. Resonant ultrasound spectroscopy is used to measure the adiabatic bulk modulus. Calorimetry is a measurement of the enthalpy and specific heat. Experimental methods also provide the temperatures of phase transitions (solid-solid, solid-liquid, liquid-gas, solid-gas). These isobaric data provide constraints to the thermal components of an EOS model. Then theoretical calculations are used for constraining our models in regions where data are often absent. For that we rely on DFT, quantum molecular dynamics, and quantum Monte Carlo calculations. These methods are computationally intensive, but better match experimental results for most materials describable by a multiphase EOS (Ref. 24). Such a modern multiphase EOS generated at LANL for aluminum is shown in Fig. 2. Other modern EOS models include that of the Lawrence Livermore National Laboratory PURGATORIO, a novel implementation of the INFERNO EOS physical model.50
Development of a combined quantum monte carlo-effective fragment molecular orbital method
Published in Molecular Physics, 2019
The Quantum Monte Carlo (QMC) method is a stochastic approach for solving the Schrodinger equation [1]. The uncertainties of the predicted QMC properties (e.g. the energy of a molecule) can be computed and controlled. Thus, the QMC results are usually very reliable, with an accuracy that is typically below 1 kcal/mol [2,3].