1st Edition

# Stochastic Simulations of Clusters Quantum Methods in Flat and Curved Spaces

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*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.

*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**

### Biography

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