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

1. Legendre Polynomials
   1.1 Potential of a Point Charge Located on the z-Axis
   1.2 Laplace’s Equation
   1.3 Solution of Laplace’s Equation in Cartesian Coordinates
   1.4 Laplace’s Equation in Spherical-Polar Coordinates
   1.5 Orthogonality and Normalization of Legendre Polynomials
   1.6 Expansion of an Arbitrary Function in Legendre Series
   1.7 Recurrence Relations for Legendre Polynomials
   1.8 Analytic Expressions for First Few Legendre Polynomials
   1.9 Symmetry Properties of Legendre Polynomials
2. Associated Legendre Functions
   2.1 Generalized Legendre Equation
   2.2 Associated Legendre Functions
   2.3 Orthogonality and Normalization of Associated Legendre Functions
   2.4 Recurrence Relations for Associated Legendre Functions
   2.5 Derivatives of Associated Legendre Functions
   2.6 Analytic Expression for First Few Associated Legendre Functions
   2.7 Symmetry Properties of Associated Legendre Functions
3. Spherical Harmonics
   3.1 Spherical Harmonics Functions
   3.2 Orthogonality and Normalization of Spherical Harmonics
   3.3 Symmetry Properties of Spherical Harmonics
   3.4 Recurrence Relations for Spherical Harmonics
   3.5 Analytic Expression for the First Few Spherical Harmonics
   3.6 Nodal Properties of Spherical Harmonics
4. Angular Momentum
   4.1 Rotation Matrices
   4.2 Unitary Matrices
   4.3 Rotation Operator
   4.4 Commutative Properties of the Angular Momentum
   4.5 Eigenvalues of the Angular Momentum
   4.6 Angular Momentum Operator in Spherical Polar Coordinates
   4.7 Eigenvectors of the Angular Momentum Operator
   4.8 Characteristic Vectors of the Rotation Operator
   4.9 Rotation of Eigenfunctions of Angular Momentum
5. Wigner Matrix
   5.1 The Euler Angles
   5.2 Wigner Matrix for j = 1
   5.3 Wigner Matrix for j = 1/2
   5.4 General Form of the Wigner Matrix Elements
   5.5 Addition Theorem for Spherical Harmonics
6. Clebsch–Gordan Coefficients
   6.1 Addition of Angular Momenta
   6.2 Evaluation of Clebsch–Gordan Coefficients
   6.3 Addition of Angular Momentum and Spin
   6.4 Rotation of the Coupled Eigenstates of Angular Momentum
7. Recurrence Relations for Wigner Matrix
   7.1 Recurrence Relations with Increment in Index m
   7.2 Recurrence Relations with Increment in Index k
8. Solid Harmonics
   8.1 Regular and Irregular Solid Harmonics
   8.2 Regular Multipole Moments
   8.3 Irregular Multipole Moments
   8.4 Computation of Electrostatic Energy via Multipole Moments
   8.5 Recurrence Relations for Regular Solid Harmonics
   8.6 Recurrence Relations for Irregular Solid Harmonics
   8.7 Generating Functions for Solid Harmonics
   8.8 Addition Theorem for Regular Solid Harmonics
   8.9 Addition Theorem for Irregular Solid Harmonics
   8.10 Transformation of the Origin of Irregular Harmonics
   8.11 Vector Diagram Approach to Multipole Translations
9. Electrostatic Force
   9.1 Gradient of Electrostatic Potential
   9.2 Differentiation of Multipole Expansion
   9.3 Differentiation of Regular Solid Harmonics in Spherical Polar Coordinates
   9.4 Differentiation of Spherical Polar Coordinates
   9.5 Differentiation of Regular Solid Harmonics in Cartesian Coordinates
   9.6 FMM Force in Cartesian Coordinates
10. Scaling of Solid Harmonics
   10.1 Optimization of Expansion of Inverse Distance Function
   10.2 Scaling of Associated Legendre Functions
   10.3 Recurrence Relations for Scaled Regular Solid Harmonics
   10.4 Recurrence Relations for Scaled Irregular Solid Harmonics
   10.5 First Few Terms of Scaled Solid Harmonics
   10.6 Design of Computer Code for Computation of Solid Harmonics
   10.7 Program Code for Computation of Multipole Expansions
   10.8 Computation of Electrostatic Force Using Scaled Solid Harmonics
   10.9 Program Code for Computation of Force
11. Scaling of Multipole Translations
   11.1 Scaling of Multipole Translation Operations
   11.2 Program Code for M2M Translation
   11.3 Program Code for M2L Translation
   11.4 Program Code for L2L Translation
12. Fast Multipole Method
   12.1 Near and Far Fields: Prerequisites for the Use of the Fast Multipole Method
   12.2 Series Convergence and Truncation of Multipole Expansion
   12.3 Hierarchical Division of Boxes in the Fast Multipole Method
   12.4 Far Field
   12.5 NF and FF Pair Counts
   12.6 FMM Algorithm
   12.7 Accuracy Assessment of Multipole Operations
13. Multipole Translations along the z-Axis
   13.1 M2M Translation along the z-Axis
   13.2 L2L Translation along the z-Axis
   13.3 M2L Translation along the z-Axis
14. Rotation of Coordinate System
   14.1 Rotation of Coordinate System to Align the z-axis with the Axis of Translation
   14.2 Rotation Matrix
   14.3 Computation of Scaled Wigner Matrix Elements with Increment in Index m
   14.4 Computation of Scaled Wigner Matrix Elements with Increment in Index k
   14.5 Program Code for Computation of Scaled Wigner Matrix Elements Based on the k-set
   14.6 Program Code for Computation of Scaled Wigner Matrix Elements Based on the m-set
15. Rotation-Based Multipole Translations
   15.1 Assembly of Rotation Matrix
   15.2 Rotation-Based M2M Operation
   15.3 Rotation-Based M2L Operation
   15.4 Rotation-Based L2L Operation
16. Periodic Boundary Condition
   16.1 Principles of Periodic Boundary Condition
   16.2 Lattice Sum for Energy in Periodic FMM
   16.3 Multipole Moments of the Central Super-Cell
   16.4 Far-Field Contribution to the Lattice Sum for Energy
   16.5 Contribution of the Near-Field Zone into the Central Unit Cell
   16.6 Derivative of Electrostatic Energy on Particles in the Central Unit Cell
   16.7 Stress Tensor
   16.8 Analytic Expression for Stress Tensor
   16.9 Lattice Sum for Stress Tensor