Introduction to the Fast Multipole Method-Topics in Computational Biophysics, Theory, and Implementation

459 Pages·2019·9.221 MB·\459

by  Victor Anisimov (Author), James J.P. Stewart (Author)

Introduction to the Fast
Multipole Method

Introduction to the Fast
Multipole Method
Topics in Computational Biophysics,
Theory, and Implementation
Victor Anisimov
James J.P. Stewart
CRC Press
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Contents
Preface ......................................................................................................................................................ix
1. Legendre Polynomials ......................................................................................................................1
1.1 Potential of a Point Charge Located on the z-Axis .................................................................1
1.2 Laplace’s Equation ..................................................................................................................3
1.3 Solution of Laplace’s Equation in Cartesian Coordinates ......................................................5
1.4 Laplace’s Equation in Spherical Polar Coordinates ...............................................................7
1.5 Orthogonality and Normalization of Legendre Polynomials ................................................11
1.6 Expansion of an Arbitrary Function in Legendre Series ......................................................15
1.7 Recurrence Relations for Legendre Polynomials ..................................................................17
1.8 Analytic Expressions for First Few Legendre Polynomials .................................................21
1.9 Symmetry Properties of Legendre Polynomials...................................................................21
2. Associated Legendre Functions ....................................................................................................27
2.1 Generalized Legendre Equation ...........................................................................................28
2.2 Associated Legendre Functions ............................................................................................32
2.3 Orthogonality and Normalization of Associated Legendre Functions ................................34
2.4 Recurrence Relations for Associated Legendre Functions ...................................................37
2.5 Derivatives of Associated Legendre Functions ....................................................................42
2.6 Analytic Expression for First Few Associated Legendre Functions ....................................44
2.7 Symmetry Properties of Associated Legendre Functions ....................................................44
3. Spherical Harmonics ......................................................................................................................51
3.1 Spherical Harmonics Functions ............................................................................................52
3.2 Orthogonality and Normalization of Spherical Harmonics .................................................53
3.3 Symmetry Properties of Spherical Harmonics .....................................................................56
3.4 Recurrence Relations for Spherical Harmonics ...................................................................59
3.5 Analytic Expression for the First Few Spherical Harmonics ...............................................63
3.6 Nodal Properties of Spherical Harmonics ............................................................................63
4. Angular Momentum ......................................................................................................................71
4.1 Rotation Matrices ..................................................................................................................72
4.2 Unitary Matrices ...................................................................................................................77
4.3 Rotation Operator .................................................................................................................82
4.4 Commutative Properties of the Angular Momentum ...........................................................86
4.5 Eigenvalues of the Angular Momentum ...............................................................................91
4.6 Angular Momentum Operator in Spherical Polar Coordinates ............................................95
4.7 Eigenvectors of the Angular Momentum Operator .............................................................101
4.8 Characteristic Vectors of the Rotation Operator ................................................................102
4.9 Rotation of Eigenfunctions of Angular Momentum ...........................................................105
5. Wigner Matrix ..............................................................................................................................109
5.1 The Euler Angles ................................................................................................................109
5.2 Wigner Matrix for j = 1 .......................................................................................................113
5.3 Wigner Matrix for j = 1/2...................................................................................................120
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5.4 General Form of the Wigner Matrix Elements ...................................................................128
5.5 Addition Theorem for Spherical Harmonics .......................................................................141
6. Clebsch–Gordan Coefficients ......................................................................................................147
6.1 Addition of Angular Momenta ............................................................................................147
6.2 Evaluation of Clebsch–Gordan Coefficients .......................................................................153
6.3 Addition of Angular Momentum and Spin .........................................................................166
6.4 Rotation of the Coupled Eigenstates of Angular Momentum .............................................171
7. Recurrence Relations for Wigner Matrix ...................................................................................177
7.1 Recurrence Relations with Increment in Index m ...............................................................177
7.2 Recurrence Relations with Increment in Index k.................................................................182
8. Solid Harmonics ............................................................................................................................189
8.1 Regular and Irregular Solid Harmonics ..............................................................................189
8.2 Regular Multipole Moments ................................................................................................191
8.3 Irregular Multipole Moments .............................................................................................192
8.4 Computation of Electrostatic Energy via Multipole Moments ...........................................193
8.5 Recurrence Relations for Regular Solid Harmonics ..........................................................194
8.6 Recurrence Relations for Irregular Solid Harmonics .........................................................196
8.7 Generating Functions for Solid Harmonics ........................................................................198
8.8 Addition Theorem for Regular Solid Harmonics ...............................................................201
8.9 Addition Theorem for Irregular Solid Harmonics ..............................................................205
8.10 Transformation of the Origin of Irregular Harmonics ........................................................210
8.11 Vector Diagram Approach to Multipole Translations .........................................................212
9. Electrostatic Force .........................................................................................................................215
9.1 Gradient of Electrostatic Potential .......................................................................................215
9.2 Differentiation of Multipole Expansion ...............................................................................217
9.3 Differentiation of Regular Solid Harmonics in Spherical Polar Coordinates .....................218
9.4 Differentiation of Spherical Polar Coordinates ..................................................................220
9.5 Differentiation of Regular Solid Harmonics in Cartesian Coordinates .............................222
9.6 FMM Force in Cartesian Coordinates ................................................................................224
10. Scaling of Solid Harmonics .........................................................................................................225
10.1 Optimization of Expansion of Inverse Distance Function .................................................225
10.2 Scaling of Associated Legendre Functions ........................................................................227
10.3 Recurrence Relations for Scaled Regular Solid Harmonics ...............................................229
10.4 Recurrence Relations for Scaled Irregular Solid Harmonics .............................................232
10.5 First Few Terms of Scaled Solid Harmonics ......................................................................235
10.6 Design of Computer Code for Computation of Solid Harmonics ......................................236
10.7 Program Code for Computation of Multipole Expansions .................................................237
10.8 Computation of Electrostatic Force Using Scaled Solid Harmonics ..................................242
10.9 Program Code for Computation of Force ...........................................................................244
11. Scaling of Multipole Translations ...............................................................................................251
11.1 Scaling of Multipole Translation Operations.......................................................................251
11.2 Program Code for M2M Translation ..................................................................................253
11.3 Program Code for M2L Translation ....................................................................................261
11.4 Program Code for L2L Translation ....................................................................................265
Contents vii
12. Fast Multipole Method .................................................................................................................275
12.1 Near and Far Fields: Prerequisites for the Use of the Fast Multipole Method ...................275
12.2 Series Convergence and Truncation of Multipole Expansion.............................................277
12.3 Hierarchical Division of Boxes in the Fast Multipole Method ...........................................284
12.4 Far Field ..............................................................................................................................288
12.5 Near Field and Far Field Pair Counts .................................................................................289
12.6 FMM Algorithm .................................................................................................................294
12.7 Accuracy Assessment of Multipole Operations ..................................................................300
13. Multipole Translations along the z-Axis ....................................................................................303
13.1 M2M Translation along the z-Axis .....................................................................................303
13.2 L2L Translation along the z-Axis ........................................................................................310
13.3 M2L Translation along the z-Axis .......................................................................................316
14. Rotation of Coordinate System ...................................................................................................325
14.1 Rotation of Coordinate System to Align the z-axis with the Axis of Translation ..............325
14.2 Rotation Matrix ...................................................................................................................328
14.3 Computation of Scaled Wigner Matrix Elements with Increment in Index m ...................330
14.4 Computation of Scaled Wigner Matrix Elements with Increment in Index k ....................337
14.5 Program Code for Computation of Scaled Wigner Matrix Elements Based
on the k-set ..........................................................................................................................342
14.6 Program Code for Computation of Scaled Wigner Matrix Elements Based
on the m-set .........................................................................................................................354
15. Rotation-Based Multipole Translations ......................................................................................361
15.1 Assembly of Rotation Matrix ..............................................................................................361
15.2 Rotation-Based M2M Operation ........................................................................................365
15.3 Rotation-Based M2L Operation .........................................................................................368
15.4 Rotation-Based L2L Operation ...........................................................................................370
16. Periodic Boundary Conditions .....................................................................................................375
16.1 Principles of Periodic Boundary Conditions .......................................................................375
16.2 Lattice Sum for Energy in Periodic FMM ..........................................................................377
16.3 Multipole Moments of the Central Super-Cell ...................................................................379
16.4 Far-Field Contribution to the Lattice Sum for Energy........................................................384
16.5 Contribution of the Near-Field Zone into the Central Unit Cell ........................................390
16.6 Derivative of Electrostatic Energy on Particles in the Central Unit Cell ............................391
16.7 Stress Tensor .......................................................................................................................392
16.8 Analytic Expression for Stress Tensor ................................................................................394
16.9 Lattice Sum for Stress Tensor .............................................................................................398
Appendix ...............................................................................................................................................407
Bibliography .........................................................................................................................................441
Index ......................................................................................................................................................443

Preface
The Fast Multipole Method (FMM) is an efficient procedure developed by Greengard and Rokhlin for
computing the potential energy in systems that obey the inverse-square law. Since the amount of work
required in computing interaction energies scales quadratically with the number of interacting particles,
the computational cost of direct summation quickly becomes prohibitive. The FMM addresses that
challenge by making the amount of work scale linearly, O(N), with system size N.
Originally developed for astrophysics, the FMM algorithm quickly gained attention as a promising
method for the treatment of electrostatic interactions in molecular modeling applications. This field
has traditionally been dominated by simpler and computationally less-demanding Particle Mesh Ewald
methods (PME), which scale as O(N log N). However, the advent of exascale computing (1018 floating
point operations per second), combined with the trend in molecular modeling to simulate larger and more
realistic systems, pushes the number of particles in the system close to a billion, and that steadily increases
the value of the true O(N) method.
Starting from the discovery of the FMM algorithm, several seminal works published by the two teams
lead by Martin Head-Gordon and Gustavo Scuseria marked significant milestones in the development of
the FMM method for biophysics applications. Although FMM theory is well understood, its complexity
and the difficulty in efficient computer implementation are two major factors that limit the adoption of the
method. Attempting to address these issues, our intent is to provide a complete, self-contained explanation
of the math of the FMM method, so that anyone having an undergraduate grasp of calculus should be
able to follow the presented material.
FMM theory can be regarded as an umbrella that covers numerous topics in mathematical physics,
and, as a result, a comprehensive introduction to FMM theory is virtually non-existent in the literature.
Despite an abundance of literature on those subjects, many publications are often either too concise
or too advanced to be used in self-study. This book responds to the need by providing a simple and
easy-to-understand introduction to every important topic, while conducting a self-contained complete
reconstruction of the FMM theory from the ground up. The reader learns the method by rederiving the
equations involved and turning them into computer code.
This project aims at a wide readership. Some, particularly those with an extensive physics background,
may find the presented material useful in the teaching process. Many, who have no time or resources to
gather the pieces of information from the distributed science publications, will be able to study the method
from a single source. Others may find complementary insights about such topics as angular momentum,
addition theorems, or Wigner matrix, which go beyond their utility for the FMM method.
Finding the best way to present all the required information in order to explain FMM requires undertaking
a separate research work. This textbook presents an attempt to logically and concisely introduce the FMM
method as if deriving it from first principles. Of necessity, many advanced topics about FMM have had
to be excluded due to space constraints. It is our hope that the material presented will give the reader the
necessary background to continue their study of the FMM theory on an independent journey.
In this book, Gauss units are employed because of their simplicity and their visual clarity in the
equations. Readers who want to should have no difficulty converting them to SI units.
Victor Anisimov
James J.P. Stewart
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