1st Edition

Stochastic Simulations of Clusters Quantum Methods in Flat and Curved Spaces

By Emanuele Curotto Copyright 2010
    696 Pages 67 B/W Illustrations
    by CRC Press

    696 Pages 67 B/W Illustrations
    by CRC Press

    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.


    FORTRAN Essentials


    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


    Intrinsic Functions

    User-Defined Functions


    Numerical Derivatives

    The Extended Trapezoid Rule to Evaluate Integrals

    Basics of Classical Dynamics


    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


    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


    A Few Useful Definitions


    Number Fields

    Vector Spaces


    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


    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


    The Failures of Classical Physics


    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


    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


    The Harmonic Oscillator

    Classical Canonical Average Energy and Heat Capacity

    Quantum Canonical Average Energy and Heat Capacity

    The Path Integral in Rd

    The Canonical Fourier Path Integral

    The Reweighted Fourier–Wiener Path Integral


    Characterization of the Potential of Ar7


    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 Ar7


    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 Ar7

    Characterization Techniques: The Lindemann Index

    Characterization Techniques: Bond Orientational Parameters

    Characterization Techniques: Structural Comparison



    Introduction to Differential Geometry


    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


    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 S2

    Variational Ground State for S2

    DMC in S2

    Stereographic Projection Path Integral in S2

    Higher Dimensional Tops

    The Free Particle in a Ring

    The Particle in a Ring Subject to Smooth Potentials


    Clusters of Rigid Tops


    The Stockmayer Model

    The Map for R3n (S2)n

    The Gradient of the Lennard-Jones Dipole-Dipole (LJDD) Potential

    Beyond the Stockmayer Model for Rigid Linear Tops

    The Hessian Metric Tensor on R3n (S2)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 R3n n

    The Hessian Metric Tensor for R3n n

    Local Energy and Action for R3n n

    Concluding Remarks




    Emanuele Curotto is a professor of chemistry at Arcadia University in Glenside, Pennsylvania.