Preface |
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ix | |
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1 | (26) |
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1.1 Potential of a Point Charge Located on the z-Axis |
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1 | (2) |
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3 | (2) |
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1.3 Solution of Laplace's Equation in Cartesian Coordinates |
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5 | (2) |
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1.4 Laplace's Equation in Spherical Polar Coordinates |
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7 | (4) |
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1.5 Orthogonality and Normalization of Legendre Polynomials |
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11 | (4) |
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1.6 Expansion of an Arbitrary Function in Legendre Series |
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15 | (2) |
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1.7 Recurrence Relations for Legendre Polynomials |
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17 | (4) |
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1.8 Analytic Expressions for First Few Legendre Polynomials |
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21 | (1) |
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1.9 Symmetry Properties of Legendre Polynomials |
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21 | (6) |
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2 Associated Legendre Functions |
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27 | (24) |
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2.1 Generalized Legendre Equation |
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28 | (4) |
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2.2 Associated Legendre Functions |
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32 | (2) |
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2.3 Orthogonality and Normalization of Associated Legendre Functions |
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34 | (3) |
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2.4 Recurrence Relations for Associated Legendre Functions |
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37 | (5) |
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2.5 Derivatives of Associated Legendre Functions |
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42 | (2) |
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2.6 Analytic Expression for First Few Associated Legendre Functions |
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44 | (1) |
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2.7 Symmetry Properties of Associated Legendre Functions |
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44 | (7) |
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51 | (20) |
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3.1 Spherical Harmonics Functions |
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52 | (1) |
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3.2 Orthogonality and Normalization of Spherical Harmonics |
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53 | (3) |
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3.3 Symmetry Properties of Spherical Harmonics |
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56 | (3) |
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3.4 Recurrence Relations for Spherical Harmonics |
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59 | (4) |
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3.5 Analytic Expression for the First Few Spherical Harmonics |
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63 | (1) |
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3.6 Nodal Properties of Spherical Harmonics |
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63 | (8) |
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71 | (38) |
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72 | (5) |
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77 | (5) |
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82 | (4) |
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4.4 Commutative Properties of the Angular Momentum |
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86 | (5) |
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4.5 Eigenvalues of the Angular Momentum |
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91 | (4) |
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4.6 Angular Momentum Operator in Spherical Polar Coordinates |
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95 | (6) |
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4.7 Eigenvectors of the Angular Momentum Operator |
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101 | (1) |
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4.8 Characteristic Vectors of the Rotation Operator |
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102 | (3) |
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4.9 Rotation of Eigenfunctions of Angular Momentum |
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105 | (4) |
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109 | (38) |
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109 | (4) |
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5.2 Wigner Matrix for j = 1 |
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113 | (7) |
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5.3 Wigner Matrix for j = 1/2 |
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120 | (8) |
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5.4 General Form of the Wigner Matrix Elements |
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128 | (13) |
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5.5 Addition Theorem for Spherical Harmonics |
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141 | (6) |
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6 Clebsch-Gordan Coefficients |
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147 | (30) |
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6.1 Addition of Angular Momenta |
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147 | (6) |
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6.2 Evaluation of Clebsch-Gordan Coefficients |
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153 | (13) |
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6.3 Addition of Angular Momentum and Spin |
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166 | (5) |
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6.4 Rotation of the Coupled Eigenstates of Angular Momentum |
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171 | (6) |
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7 Recurrence Relations for Wigner Matrix |
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177 | (12) |
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7.1 Recurrence Relations with Increment in Index m |
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177 | (5) |
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7.2 Recurrence Relations with Increment in Index k |
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182 | (7) |
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189 | (26) |
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8.1 Regular and Irregular Solid Harmonics |
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189 | (2) |
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8.2 Regular Multipole Moments |
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191 | (1) |
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8.3 Irregular Multipole Moments |
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192 | (1) |
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8.4 Computation of Electrostatic Energy via Multipole Moments |
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193 | (1) |
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8.5 Recurrence Relations for Regular Solid Harmonics |
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194 | (2) |
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8.6 Recurrence Relations for Irregular Solid Harmonics |
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196 | (2) |
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8.7 Generating Functions for Solid Harmonics |
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198 | (3) |
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8.8 Addition Theorem for Regular Solid Harmonics |
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201 | (4) |
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8.9 Addition Theorem for Irregular Solid Harmonics |
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205 | (5) |
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8.10 Transformation of the Origin of Irregular Harmonics |
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210 | (2) |
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8.11 Vector Diagram Approach to Multipole Translations |
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212 | (3) |
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215 | (10) |
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9.1 Gradient of Electrostatic Potential |
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215 | (2) |
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9.2 Differentiation of Multipole Expansion |
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217 | (1) |
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9.3 Differentiation of Regular Solid Harmonics in Spherical Polar Coordinates |
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218 | (2) |
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9.4 Differentiation of Spherical Polar Coordinates |
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220 | (2) |
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9.5 Differentiation of Regular Solid Harmonics in Cartesian Coordinates |
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222 | (2) |
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9.6 FMM Force in Cartesian Coordinates |
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224 | (1) |
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10 Scaling of Solid Harmonics |
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225 | (26) |
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10.1 Optimization of Expansion of Inverse Distance Function |
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225 | (2) |
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10.2 Scaling of Associated Legendre Functions |
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227 | (2) |
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10.3 Recurrence Relations for Scaled Regular Solid Harmonics |
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229 | (3) |
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10.4 Recurrence Relations for Scaled Irregular Solid Harmonics |
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232 | (3) |
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10.5 First Few Terms of Scaled Solid Harmonics |
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235 | (1) |
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10.6 Design of Computer Code for Computation of Solid Harmonics |
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236 | (1) |
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10.7 Program Code for Computation of Multipole Expansions |
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237 | (5) |
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10.8 Computation of Electrostatic Force Using Scaled Solid Harmonics |
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242 | (2) |
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10.9 Program Code for Computation of Force |
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244 | (7) |
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11 Scaling of Multipole Translations |
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251 | (24) |
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11.1 Scaling of Multipole Translation Operations |
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251 | (2) |
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11.2 Program Code for M2M Translation |
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253 | (8) |
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11.3 Program Code for M2L Translation |
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261 | (4) |
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11.4 Program Code for L2L Translation |
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265 | (10) |
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275 | (28) |
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12.1 Near and Far Fields: Prerequisites for the Use of the Fast Multipole Method |
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275 | (2) |
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12.2 Series Convergence and Truncation of Multipole Expansion |
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277 | (7) |
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12.3 Hierarchical Division of Boxes in the Fast Multipole Method |
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284 | (4) |
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288 | (1) |
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12.5 Near Field and Far Field Pair Counts |
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289 | (5) |
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294 | (6) |
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12.7 Accuracy Assessment of Multipole Operations |
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300 | (3) |
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13 Multipole Translations along the z-Axis |
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303 | (22) |
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13.1 M2M Translation along the z-Axis |
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303 | (7) |
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13.2 L2L Translation along the z-Axis |
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310 | (6) |
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13.3 M2L Translation along the z-Axis |
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316 | (9) |
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14 Rotation of Coordinate System |
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325 | (36) |
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14.1 Rotation of Coordinate System to Align the z-axis with the Axis of Translation |
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325 | (3) |
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328 | (2) |
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14.3 Computation of Scaled Wigner Matrix Elements with Increment in Index m |
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330 | (7) |
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14.4 Computation of Scaled Wigner Matrix Elements with Increment in Index k |
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337 | (5) |
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14.5 Program Code for Computation of Scaled Wigner Matrix Elements Based on the k-set |
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342 | (12) |
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14.6 Program Code for Computation of Scaled Wigner Matrix Elements Based on the m-set |
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354 | (7) |
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15 Rotation-Based Multipole Translations |
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361 | (14) |
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15.1 Assembly of Rotation Matrix |
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361 | (4) |
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15.2 Rotation-Based M2M Operation |
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365 | (3) |
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15.3 Rotation-Based M2L Operation |
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368 | (2) |
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15.4 Rotation-Based L2L Operation |
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370 | (5) |
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16 Periodic Boundary Conditions |
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375 | (32) |
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16.1 Principles of Periodic Boundary Conditions |
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375 | (2) |
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16.2 Lattice Sum for Energy in Periodic FMM |
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377 | (2) |
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16.3 Multipole Moments of the Central Super-Cell |
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379 | (5) |
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16.4 Far-Field Contribution to the Lattice Sum for Energy |
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384 | (6) |
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16.5 Contribution of the Near-Field Zone into the Central Unit Cell |
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390 | (1) |
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16.6 Derivative of Electrostatic Energy on Particles in the Central Unit Cell |
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391 | (1) |
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392 | (2) |
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16.8 Analytic Expression for Stress Tensor |
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394 | (4) |
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16.9 Lattice Sum for Stress Tensor |
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398 | (9) |
Appendix |
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407 | (34) |
Bibliography |
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441 | (2) |
Index |
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443 | |