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This book introduces the reader to the theory and computer implementation of the Fast Multipole Method. It covers the topics of Laplace’s equation, spherical harmonics, angular momentum, the Wigner matrix, the addition theorem for solid harmonics, and lattice sums for periodic boundary conditions, along with providing a complete, self-contained explanation of the math of the method, so that anyone having an undergraduate grasp of calculus should be able to follow the material presented. The authors derive the Fast Multipole Method from first principles and systematically construct the theory connecting all the parts.

Key Features:

- Introduces each topic from first principles
- Derives every equation presented, and explains each step in its derivation
- Builds the necessary theory in order to understand, develop, and use the method
- Describes the conversion from theory to computer implementation
- Guides through code optimization and parallelization

*Autho Biography*

Victor Anisimov is an Application Performance Engineer at the Argonne National Laboratory

James J. P. Stewart is the author of the MOPAC program developed by Stewart Computational Chemistry, LLC

**Introduction to Biomolecular Simulations**

Dawn of Quantum Biology

What Makes Biomolecules Special

Matching Computational and Experimental Conditions

**Mathematical Preliminaries**

Linear Algebra

Spherical Coordinates

Internal Coordinates

Coordinate Transformation

Differential and Integral Calculus

Dirac Delta Function

Linear Operators

Function Minimization

Mathematical Statistics

**Classical Mechanics**

The Newton Laws

The Lagrange Equation

**Classical Electrostatics**

Electrostatic Potential

Electric Field

Curl and Divergence

Gauss' Law

Poisson and Laplace Equations

The Legendre Polinomials

Spherical Harmonics

Multipole Expansion

Fourier Series

**Many-Electron Systems**

The Schrodinger Equation

The Born-Oppenheimer Approximation

Variation Principle

Perturbation Theory

The Hartree-Fock Approximation

Geometry Optimization

Constrained Optimization using Lagrange Multipliers

Coupled Perturbed Hartree-Fock Theory

Vibrational Analysis

**The Semiempirical Theory**

The NDDO Approximation

Energy Derivatives

Linear Scaling Methods

Initial Guess

VFL Approximation and LocalSCF Method

Multiple-Layer QM Models

Implicit Solvent Model COSMO

**Classical Biomolecular Force Fields**

Mean Field Approximation

Simulation of Bulk Liquids

Determination of Electrostatic and Lennard-Jones Parameters

Biomolecular Force Fields and Programs

**Long-Range Electrostatics**

Fast Multipole Method

Ewald Sums Method

**Condensed Phase**

Translational Symmetry

Periodic Structures

Hartree-Fock Method for Periodic Systems

**Statistical Mechanics**

Statistical Ensembles

Configuration Space

Entropy

Time-Correlation Function

Radial Distribution Function

**Molecular Dynamics**

Hamiltonian Dynamics

Langevin Dynamics

Non-Hamiltonian Dynamics

Constrained Dynamics

**Free Energy Calculations**

Free Energy Problem

Potential of Mean Force

Free Energy Pe11urbation

Thermodynamic Integration

The Weighted Histogram Analysis Method

**Sample Applications**

Structure Preconditioning

Water Solvation

Adding Ions

Structure Relaxation by Using Force Fields

Quantum Mechanical Molecular Dynamics

Chemical Reactions

Conformational Sampling

Protein-Ligand Binding Free Energy

**High Performance Computing**

MPI Basics

Submitting Multiple Jobs on Multiple Processors