Unravels Complex Problems through Quantum Monte Carlo Methods
Clusters hold the key to our understanding of intermolecular forces and how these affect the physical properties of bulk condensed matter. They can be found in a multitude of important applications, including novel fuel materials, atmospheric chemistry, semiconductors, nanotechnology, and computational biology. Focusing on the class of weakly bound substances known as van derWaals clusters or complexes, Stochastic Simulations of Clusters: Quantum Methods in Flat and Curved Spaces presents advanced quantum simulation techniques for condensed matter.
The book develops finite temperature statistical simulation tools and real-time algorithms for the exact solution of the Schrödinger equation. It draws on potential energy models to gain insight into the behavior of minima and transition states. Using Monte Carlo methods as well as ground state variational and diffusion Monte Carlo (DMC) simulations, the author explains how to obtain temperature and quantum effects. He also shows how the path integral approach enables the study of quantum effects at finite temperatures.
To overcome timescale problems, this book supplies efficient and accurate methods, such as diagonalization techniques, differential geometry, the path integral method in statistical mechanics, and the DMC approach. Gleaning valuable information from recent research in this area, it presents special techniques for accelerating the convergence of quantum Monte Carlo methods.
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More About This Textbook
Overview
Unravels Complex Problems through Quantum Monte Carlo Methods
Clusters hold the key to our understanding of intermolecular forces and how these affect the physical properties of bulk condensed matter. They can be found in a multitude of important applications, including novel fuel materials, atmospheric chemistry, semiconductors, nanotechnology, and computational biology. Focusing on the class of weakly bound substances known as van derWaals clusters or complexes, Stochastic Simulations of Clusters: Quantum Methods in Flat and Curved Spaces presents advanced quantum simulation techniques for condensed matter.
The book develops finite temperature statistical simulation tools and real-time algorithms for the exact solution of the Schrödinger equation. It draws on potential energy models to gain insight into the behavior of minima and transition states. Using Monte Carlo methods as well as ground state variational and diffusion Monte Carlo (DMC) simulations, the author explains how to obtain temperature and quantum effects. He also shows how the path integral approach enables the study of quantum effects at finite temperatures.
To overcome timescale problems, this book supplies efficient and accurate methods, such as diagonalization techniques, differential geometry, the path integral method in statistical mechanics, and the DMC approach. Gleaning valuable information from recent research in this area, it presents special techniques for accelerating the convergence of quantum Monte Carlo methods.
Product Details
Related Subjects
Meet the Author
Emanuele Curotto is a professor of chemistry at Arcadia University in Glenside, Pennsylvania.
Table of Contents
FUNDAMENTALS
FORTRAN Essentials
Introduction
What Is FORTRAN?
FORTRAN Basics
Data Types
The IMPLICIT Statement
Initialization and Assignment
Order of Operations
Column Position Rules
A Typical Chemistry Problem Solved with FORTRAN
Free Format I/O
The FORTRAN Code for the Tertiary Mixtures Problem
Conditional Execution
Loops
Intrinsic Functions
User-Defined Functions
Subroutines
Numerical Derivatives
The Extended Trapezoid Rule to Evaluate Integrals
Basics of Classical Dynamics
Introduction
Some Important Variables of Classical Physics
The Lagrangian and the Hamiltonian in Euclidean Spaces
The Least Action Principle and the Equations of Motion
The Two-Body Problem with Central Potential
Isotropic Potentials and the Two-Body Problem
The Rigid Rotor
Numerical Integration Methods
Hamilton’s Equations and Symplectic Integrators
The Potential Energy Surface
Dissipative Systems
The Fourier Transform and the Position Autocorrelation Function
Basics of Stochastic Computations
Introduction
Continuous Random Variables and Their Distributions
The Classical Statistical Mechanics of a Single Particle
The Monoatomic Ideal Gas
The Equipartition Theorem
Basics of Stochastic Computation
Probability Distributions
Minimizing V by Trial and Error
The Metropolis Algorithm
Parallel Tempering
A Random Number Generator
Vector Spaces, Groups, and Algebras
Introduction
A Few Useful Definitions
Groups
Number Fields
Vector Spaces
Algebras
The Exponential Mapping of Lie Algebras
The Determinant of a n × n Matrix and the Levi–Civita Symbol
Scalar Product, Outer Product, and Vector Space Mapping
Rotations in Euclidean Space
Complex Field Extensions
Dirac Bra–Ket Notation
Eigensystems
The Connection between Diagonalization and Lie Algebras
Symplectic Lie Algebras and Groups
Lie Groups as Solutions of Differential Equations
Split Symplectic Integrators
Supermatrices and Superalgebras
Matrix Quantum Mechanics
Introduction
The Failures of Classical Physics
Spectroscopy
The Heat Capacity of Solids at Low Temperature
The Photoelectric Effect
Black Body Radiator
The Beginning of the Quantum Revolution
Modern Quantum Theory and Schrödinger’s Equation
Matrix Quantum Mechanics
The Monodimensional Hamiltonian in a Simple Hilbert Space
Numerical Solution Issues in Vector Spaces
The Harmonic Oscillator in Hilbert Space
A Simple Discrete Variable Representation (DVR)
Accelerating the Convergence of the Simple DVR
Elements of Sparse Matrix Technology
The Gram–Schmidt Process
The Krylov Space
The Row Representation of a Sparse Matrix
The Lanczos Algorithm
Orbital Angular Momentum and the Spherical Harmonics
Complete Sets of Commuting Observables
The Addition of Angular Momentum Vectors
Computation of the Vector Coupling Coefficients
Matrix Elements of Anisotropic Potentials in the Angular Momentum Basis
The Physical Rigid Dipole in a Constant Electric Field
Time Evolution in Quantum Mechanics
Introduction
The Time-Dependent Schrödinger Equation
Wavepackets, Measurements, and Time Propagation of Wavepackets
The Time Evolution Operator
The Dyson Series and the Time-Ordered Exponential Representation
The Magnus Expansion
The Trotter Factorization
The time_evolution_operator Program
Feynman’s Path Integral
Quantum Monte Carlo
A Variational Monte Carlo Method for Importance Sampling Diffusion Monte Carlo (IS-DMC)
IS-DMC with Drift
Green’s Function Diffusion Monte Carlo
The Path Integral in Euclidean Spaces
Introduction
The Harmonic Oscillator
Classical Canonical Average Energy and Heat Capacity
Quantum Canonical Average Energy and Heat Capacity
The Path Integral in R^{d}
The Canonical Fourier Path Integral
The Reweighted Fourier–Wiener Path Integral
ATOMIC CLUSTERS
Characterization of the Potential of Ar_{7}
Introduction
Cartesian Coordinates of Atomic Clusters
Rotations and Translations
The Center of Mass
The Inertia Tensor
The Structural Comparison Algorithm
Gradients and Hessians of Multidimensional Potentials
The Lennard–Jones Potential V^{(LJ)}
The Gradient of V^{(LJ)}
Brownian Dynamics at 0 K
Basin Hopping
The Genetic Algorithm
The Hessian Matrix
Normal Mode Analysis
Transition States with the Cerjan–Miller Algorithm
Optical Activity
Classical and Quantum Simulations of Ar_{7}
Introduction
Simulation Techniques: Parallel Tempering Revisited
Thermodynamic Properties of a Cluster with n Atoms
The Program parallel_tempering_r3n.f
The Variational Ground State Energy
Diffusion Monte Carlo (DMC) of Atomic Clusters
Path Integral Simulations of Ar_{7}
Characterization Techniques: The Lindemann Index
Characterization Techniques: Bond Orientational Parameters
Characterization Techniques: Structural Comparison
Appendices
METHODS IN CURVED SPACES
Introduction to Differential Geometry
Introduction
Coordinate Changes: Einstein’s Sum Convention and the Metric Tensor
Contravariant Tensors
Gradients as 1-Forms
Tensors of Higher Ranks
The Metric Tensor of a Space
Integration on Manifolds
Stereographic Projections
Dynamics in Manifolds
The Hessian Metric
The Christofell Connections and the Geodesic Equations
The Laplace–Beltrami Operator
The Riemann–Cartan Curvature Scalar
The Two-Body Problem Revisited
Stereographic Projections for the Two-Body Problem
The Rigid Rotor and the Infinitely Stiff Spring Constant Limit
Relative Coordinates for the Three-Body Problem
The Rigid-Body Problem and the Body Fixed Frame
Stereographic Projections for the Ellipsoid of Inertia
The Spherical Top
The Riemann Curvature Scalar for a Spherical Top
Coefficients and the Curvature for Spherical Tops with Stereographic Projection Coordinates (SPCs)
The Riemann Curvature Scalar for a Symmetric Nonspherical Top
A Split Operator for Symplectic Integrators in Curved Manifolds
The Verlet Algorithm for Manifolds
Simulations in Curved Manifolds
Introduction
The Invariance of the Phase Space Volume
Variational Ground States
DMC in Manifolds
The Path Integral in Space-Like Curved Manifolds
The Virial Estimator for the Total Energy
Angular Momentum Theory Solution for a Particle in S^{2}
Variational Ground State for S^{2}
DMC in S^{2}
Stereographic Projection Path Integral in S^{2}
Higher Dimensional Tops
The Free Particle in a Ring
The Particle in a Ring Subject to Smooth Potentials
APPLICATIONS TO MOLECULAR SYSTEMS
Clusters of Rigid Tops
Introduction
The Stockmayer Model
The Map for R^{3n} (S^{2})^{n}
The Gradient of the Lennard-Jones Dipole-Dipole (LJDD) Potential
Beyond the Stockmayer Model for Rigid Linear Tops
The Hessian Metric Tensor on R^{3n} (S^{2})^{n}
Reweighted Random Series Action for Clusters of Linear Rigid Tops
The Local Energy Estimator for Clusters of Linear Rigid Tops
Clusters of Rigid Nonlinear Tops
Coordinate Transformations for R^{3n} ^{n}
The Hessian Metric Tensor for R^{3n} ^{n}
Local Energy and Action for R^{3n} ^{n}
Concluding Remarks
Bibliography
Index