Thomson problem

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The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons on the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904[1] after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.

Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy.

Mathematical statement

The physical system embodied by the Thomson problem is a special case of one of eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere".[2] The solution of each N-electron problem is obtained when the N-electron configuration constrained to the surface of a sphere of unit radius, r=1, yields a global electrostatic potential energy minimum, U(N).

The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's Law,

U_{ij}(N)=k_e{e_i e_j \over r_{ij}}.

Here, k_e is Coulomb's constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively.

Simplified units of e=1 and k_e=1 are used without loss of generality. Then,

U_{ij}(N) = {1 \over r_{ij}}.

The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interactions

U(N) = \sum_{i < j} \frac{1}{r_{ij}}.

The global minimization of U(N) over all possible collections of N distinct points is typically found by numerical minimization algorithms.

Example

The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin, r_{ij} = 2r = 2, or

U(2) = {1 \over 2}.

Known solutions

Minimum energy configurations have been rigorously identified in only a handful of cases.

  • For N=1, the solution is trivial as the electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero as the electron is not subject to the electric field due to any other sources of charge.
  • For N=3, electrons reside at the vertices of an equilateral triangle about a great circle.[3]
  • For N=4, electrons reside at the vertices of a regular tetrahedron.
  • For N=5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid.[4]
  • For N=6, electrons reside at vertices of a regular octahedron.[5]
  • For N=12, electrons reside at the vertices of a regular icosahedron.[6]

Notably, the geometric solutions of the Thomson problem for N=4, 6, and 12 electrons are known as Platonic solids whose faces are all congruent equilateral triangles. Numerical solutions for N=8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square.

Generalizations

One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional \sum_{i < j} f(|x_i-x_j|)

Traditionally, one considers f(x)=x^{-\alpha} . Notable cases include α = ∞, the Tammes problem (packing); α = 1, the Thomson problem; α = 0, Whyte's problem (to maximize the product of distances).

One may also consider configurations of N points on a sphere of higher dimension. See spherical design.

Relations to other scientific problems

The Thomson problem is a natural consequence of Thomson's plum pudding model in the absence of its uniform positive background charge.[7]

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"No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."

—Sir J. J. Thomson[8]

Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements.[9]

The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps.

The generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR Theory. An example with long-range logarithmic interactions is provided by the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.

Configurations of smallest known energy

In the following table N is the number of points (charges) in a configuration, E_1 is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and r_i are the positions of the charges. Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero.

It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, v_i is the number of vertices where the given number of edges meet, e is the total number of edges, f_3 is the number of triangular faces, f_4 is the number of quadrilateral faces, and \theta_1 is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal; thus (except in the cases N=4,6,12) the convex hull is only topologically equivalent to the uniform polyhedron or Johnson solid listed in the last column.[10]

N E_{1} Symmetry \left| \sum \mathbf{r}_{i} \right| v_{3} v_{4} v_{5} v_{6} v_{7} v_{8} e f_{3} f_{4} \theta_{1} Equivalent Polyhedron
2 0.500000000 D_{\infty h} 0 – – – – – – – – – 180.000°
3 1.732050808 D_{3h} 0 – – – – – – – – – 120.000°
4 3.674234614 T_{d} 0 4 0 0 0 0 0 6 4 0 109.471° tetrahedron
5 6.474691495 D_{3h} 0 2 3 0 0 0 0 9 6 0 90.000° triangular dipyramid
6 9.985281374 O_{h} 0 0 6 0 0 0 0 12 8 0 90.000° octahedron
7 14.452977414 D_{5h} 0 0 5 2 0 0 0 15 10 0 72.000° pentagonal dipyramid
8 19.675287861 D_{4d} 0 0 8 0 0 0 0 16 8 2 71.694° square antiprism
9 25.759986531 D_{3h} 0 0 3 6 0 0 0 21 14 0 61.190° triaugmented triangular prism
10 32.716949460 D_{4d} 0 0 2 8 0 0 0 24 16 0 64.996° gyroelongated square dipyramid
11 40.596450510 C_{2v} 0.013219635 0 2 8 1 0 0 27 18 0 58.540°
12 49.165253058 I_{h} 0 0 0 12 0 0 0 30 20 0 63.435° icosahedron
13 58.853230612 C_{2v} 0.008820367 0 1 10 2 0 0 33 22 0 52.317°
14 69.306363297 D_{6d} 0 0 0 12 2 0 0 36 24 0 52.866° gyroelongated hexagonal dipyramid
15 80.670244114 D_{3} 0 0 0 12 3 0 0 39 26 0 49.225°
16 92.911655302 T 0 0 0 12 4 0 0 42 28 0 48.936°
17 106.050404829 D_{5h} 0 0 0 12 5 0 0 45 30 0 50.108°
18 120.084467447 D_{4d} 0 0 2 8 8 0 0 48 32 0 47.534°
19 135.089467557 C_{2v} 0.000135163 0 0 14 5 0 0 50 32 1 44.910°
20 150.881568334 D_{3h} 0 0 0 12 8 0 0 54 36 0 46.093°
21 167.641622399 C_{2v} 0.001406124 0 1 10 10 0 0 57 38 0 44.321°
22 185.287536149 T_{d} 0 0 0 12 10 0 0 60 40 0 43.302°
23 203.930190663 D_{3} 0 0 0 12 11 0 0 63 42 0 41.481°
24 223.347074052 O 0 0 0 24 0 0 0 60 32 6 42.065° snub cube
25 243.812760299 C_{s} 0.001021305 0 0 14 11 0 0 68 44 1 39.610°
26 265.133326317 C_{2} 0.001919065 0 0 12 14 0 0 72 48 0 38.842°
27 287.302615033 D_{5h} 0 0 0 12 15 0 0 75 50 0 39.940°
28 310.491542358 T 0 0 0 12 16 0 0 78 52 0 37.824°
29 334.634439920 D_{3} 0 0 0 12 17 0 0 81 54 0 36.391°
30 359.603945904 D_{2} 0 0 0 12 18 0 0 84 56 0 36.942°
31 385.530838063 C_{3v} 0.003204712 0 0 12 19 0 0 87 58 0 36.373°
32 412.261274651 I_{h} 0 0 0 12 20 0 0 90 60 0 37.377°
33 440.204057448 C_{s} 0.004356481 0 0 15 17 1 0 92 60 1 33.700°
34 468.904853281 D_{2} 0 0 0 12 22 0 0 96 64 0 33.273°
35 498.569872491 C_{2} 0.000419208 0 0 12 23 0 0 99 66 0 33.100°
36 529.122408375 D_{2} 0 0 0 12 24 0 0 102 68 0 33.229°
37 560.618887731 D_{5h} 0 0 0 12 25 0 0 105 70 0 32.332°
38 593.038503566 D_{6d} 0 0 0 12 26 0 0 108 72 0 33.236°
39 626.389009017 D_{3h} 0 0 0 12 27 0 0 111 74 0 32.053°
40 660.675278835 T_{d} 0 0 0 12 28 0 0 114 76 0 31.916°
41 695.916744342 D_{3h} 0 0 0 12 29 0 0 117 78 0 31.528°
42 732.078107544 D_{5h} 0 0 0 12 30 0 0 120 80 0 31.245°
43 769.190846459 C_{2v} 0.000399668 0 0 12 31 0 0 123 82 0 30.867°
44 807.174263085 O_{h} 0 0 0 24 20 0 0 120 72 6 31.258°
45 846.188401061 D_{3} 0 0 0 12 33 0 0 129 86 0 30.207°
46 886.167113639 T 0 0 0 12 34 0 0 132 88 0 29.790°
47 927.059270680 C_{s} 0.002482914 0 0 14 33 0 0 134 88 1 28.787°
48 968.713455344 O 0 0 0 24 24 0 0 132 80 6 29.690°
49 1011.557182654 C_{3} 0.001529341 0 0 12 37 0 0 141 94 0 28.387°
50 1055.182314726 D_{6d} 0 0 0 12 38 0 0 144 96 0 29.231°
51 1099.819290319 D_{3} 0 0 0 12 39 0 0 147 98 0 28.165°
52 1145.418964319 C_{3} 0.000457327 0 0 12 40 0 0 150 100 0 27.670°
53 1191.922290416 C_{2v} 0.000278469 0 0 18 35 0 0 150 96 3 27.137°
54 1239.361474729 C_{2} 0.000137870 0 0 12 42 0 0 156 104 0 27.030°
55 1287.772720783 C_{2} 0.000391696 0 0 12 43 0 0 159 106 0 26.615°
56 1337.094945276 D_{2} 0 0 0 12 44 0 0 162 108 0 26.683°
57 1387.383229253 D_{3} 0 0 0 12 45 0 0 165 110 0 26.702°
58 1438.618250640 D_{2} 0 0 0 12 46 0 0 168 112 0 26.155°
59 1490.773335279 C_{2} 0.000154286 0 0 14 43 2 0 171 114 0 26.170°
60 1543.830400976 D_{3} 0 0 0 12 48 0 0 174 116 0 25.958°
61 1597.941830199 C_{1} 0.001091717 0 0 12 49 0 0 177 118 0 25.392°
62 1652.909409898 D_{5} 0 0 0 12 50 0 0 180 120 0 25.880°
63 1708.879681503 D_{3} 0 0 0 12 51 0 0 183 122 0 25.257°
64 1765.802577927 D_{2} 0 0 0 12 52 0 0 186 124 0 24.920°
65 1823.667960264 C_{2} 0.000399515 0 0 12 53 0 0 189 126 0 24.527°
66 1882.441525304 C_{2} 0.000776245 0 0 12 54 0 0 192 128 0 24.765°
67 1942.122700406 D_{5} 0 0 0 12 55 0 0 195 130 0 24.727°
68 2002.874701749 D_{2} 0 0 0 12 56 0 0 198 132 0 24.433°
69 2064.533483235 D_{3} 0 0 0 12 57 0 0 201 134 0 24.137°
70 2127.100901551 D_{2d} 0 0 0 12 50 0 0 200 128 4 24.291°
71 2190.649906425 C_{2} 0.001256769 0 0 14 55 2 0 207 138 0 23.803°
72 2255.001190975 I 0 0 0 12 60 0 0 210 140 0 24.492°
73 2320.633883745 C_{2} 0.001572959 0 0 12 61 0 0 213 142 0 22.810°
74 2387.072981838 C_{2} 0.000641539 0 0 12 62 0 0 216 144 0 22.966°
75 2454.369689040 D_{3} 0 0 0 12 63 0 0 219 146 0 22.736°
76 2522.674871841 C_{2} 0.000943474 0 0 12 64 0 0 222 148 0 22.886°
77 2591.850152354 D_{5} 0 0 0 12 65 0 0 225 150 0 23.286°
78 2662.046474566 T_{h} 0 0 0 12 66 0 0 228 152 0 23.426°
79 2733.248357479 C_{s} 0.000702921 0 0 12 63 1 0 230 152 1 22.636°
80 2805.355875981 D_{4d} 0 0 0 16 64 0 0 232 152 2 22.778°
81 2878.522829664 C_{2} 0.000194289 0 0 12 69 0 0 237 158 0 21.892°
82 2952.569675286 D_{2} 0 0 0 12 70 0 0 240 160 0 22.206°
83 3027.528488921 C_{2} 0.000339815 0 0 14 67 2 0 243 162 0 21.646°
84 3103.465124431 C_{2} 0.000401973 0 0 12 72 0 0 246 164 0 21.513°
85 3180.361442939 C_{2} 0.000416581 0 0 12 73 0 0 249 166 0 21.498°
86 3258.211605713 C_{2} 0.001378932 0 0 12 74 0 0 252 168 0 21.522°
87 3337.000750014 C_{2} 0.000754863 0 0 12 75 0 0 255 170 0 21.456°
88 3416.720196758 D_{2} 0 0 0 12 76 0 0 258 172 0 21.486°
89 3497.439018625 C_{2} 0.000070891 0 0 12 77 0 0 261 174 0 21.182°
90 3579.091222723 D_{3} 0 0 0 12 78 0 0 264 176 0 21.230°
91 3661.713699320 C_{2} 0.000033221 0 0 12 79 0 0 267 178 0 21.105°
92 3745.291636241 D_{2} 0 0 0 12 80 0 0 270 180 0 21.026°
93 3829.844338421 C_{2} 0.000213246 0 0 12 81 0 0 273 182 0 20.751°
94 3915.309269620 D_{2} 0 0 0 12 82 0 0 276 184 0 20.952°
95 4001.771675565 C_{2} 0.000116638 0 0 12 83 0 0 279 186 0 20.711°
96 4089.154010060 C_{2} 0.000036310 0 0 12 84 0 0 282 188 0 20.687°
97 4177.533599622 C_{2} 0.000096437 0 0 12 85 0 0 285 190 0 20.450°
98 4266.822464156 C_{2} 0.000112916 0 0 12 86 0 0 288 192 0 20.422°
99 4357.139163132 C_{2} 0.000156508 0 0 12 87 0 0 291 194 0 20.284°
100 4448.350634331 T 0 0 0 12 88 0 0 294 196 0 20.297°
101 4540.590051694 D_{3} 0 0 0 12 89 0 0 297 198 0 20.011°
102 4633.736565899 D_{3} 0 0 0 12 90 0 0 300 200 0 20.040°
103 4727.836616833 C_{2} 0.000201245 0 0 12 91 0 0 303 202 0 19.907°
104 4822.876522746 D_{6} 0 0 0 12 92 0 0 306 204 0 19.957°
105 4919.000637616 D_{3} 0 0 0 12 93 0 0 309 206 0 19.842°
106 5015.984595705 D_{2} 0 0 0 12 94 0 0 312 208 0 19.658°
107 5113.953547724 C_{2} 0.000064137 0 0 12 95 0 0 315 210 0 19.327°
108 5212.813507831 C_{2} 0.000432525 0 0 12 96 0 0 318 212 0 19.327°
109 5312.735079920 C_{2} 0.000647299 0 0 14 93 2 0 321 214 0 19.103°
110 5413.549294192 D_{6} 0 0 0 12 98 0 0 324 216 0 19.476°
111 5515.293214587 D_{3} 0 0 0 12 99 0 0 327 218 0 19.255°
112 5618.044882327 D_{5} 0 0 0 12 100 0 0 330 220 0 19.351°
113 5721.824978027 D_{3} 0 0 0 12 101 0 0 333 222 0 18.978°
114 5826.521572163 C_{2} 0.000149772 0 0 12 102 0 0 336 224 0 18.836°
115 5932.181285777 C_{3} 0.000049972 0 0 12 103 0 0 339 226 0 18.458°
116 6038.815593579 C_{2} 0.000259726 0 0 12 104 0 0 342 228 0 18.386°
117 6146.342446579 C_{2} 0.000127609 0 0 12 105 0 0 345 230 0 18.566°
118 6254.877027790 C_{2} 0.000332475 0 0 12 106 0 0 348 232 0 18.455°
119 6364.347317479 C_{2} 0.000685590 0 0 12 107 0 0 351 234 0 18.336°
120 6474.756324980 C_{s} 0.001373062 0 0 12 108 0 0 354 236 0 18.418°
121 6586.121949584 C_{3} 0.000838863 0 0 12 109 0 0 357 238 0 18.199°
122 6698.374499261 I_{h} 0 0 0 12 110 0 0 360 240 0 18.612°
123 6811.827228174 C_{2v} 0.001939754 0 0 14 107 2 0 363 242 0 17.840°
124 6926.169974193 D_{2} 0 0 0 12 112 0 0 366 244 0 18.111°
125 7041.473264023 C_{2} 0.000088274 0 0 12 113 0 0 369 246 0 17.867°
126 7157.669224867 D_{4} 0 0 2 16 100 8 0 372 248 0 17.920°
127 7274.819504675 D_{5} 0 0 0 12 115 0 0 375 250 0 17.877°
128 7393.007443068 C_{2} 0.000054132 0 0 12 116 0 0 378 252 0 17.814°
129 7512.107319268 C_{2} 0.000030099 0 0 12 117 0 0 381 254 0 17.743°
130 7632.167378912 C_{2} 0.000025622 0 0 12 118 0 0 384 256 0 17.683°
131 7753.205166941 C_{2} 0.000305133 0 0 12 119 0 0 387 258 0 17.511°
132 7875.045342797 I 0 0 0 12 120 0 0 390 260 0 17.958°
133 7998.179212898 C_{3} 0.000591438 0 0 12 121 0 0 393 262 0 17.133°
134 8122.089721194 C_{2} 0.000470268 0 0 12 122 0 0 396 264 0 17.214°
135 8246.909486992 D_{3} 0 0 0 12 123 0 0 399 266 0 17.431°
136 8372.743302539 T 0 0 0 12 124 0 0 402 268 0 17.485°
137 8499.534494782 D_{5} 0 0 0 12 125 0 0 405 270 0 17.560°
138 8627.406389880 C_{2} 0.000473576 0 0 12 126 0 0 408 272 0 16.924°
139 8756.227056057 C_{2} 0.000404228 0 0 12 127 0 0 411 274 0 16.673°
140 8885.980609041 C_{1} 0.000630351 0 0 13 126 1 0 414 276 0 16.773°
141 9016.615349190 C_{2v} 0.000376365 0 0 14 126 0 1 417 278 0 16.962°
142 9148.271579993 C_{2} 0.000550138 0 0 12 130 0 0 420 280 0 16.840°
143 9280.839851192 C_{2} 0.000255449 0 0 12 131 0 0 423 282 0 16.782°
144 9414.371794460 D_{2} 0 0 0 12 132 0 0 426 284 0 16.953°
145 9548.928837232 C_{s} 0.000094938 0 0 12 133 0 0 429 286 0 16.841°
146 9684.381825575 D_{2} 0 0 0 12 134 0 0 432 288 0 16.905°
147 9820.932378373 C_{2} 0.000636651 0 0 12 135 0 0 435 290 0 16.458°
148 9958.406004270 C_{2} 0.000203701 0 0 12 136 0 0 438 292 0 16.627°
149 10096.859907397 C_{1} 0.000638186 0 0 14 133 2 0 441 294 0 16.344°
150 10236.196436701 T 0 0 0 12 138 0 0 444 296 0 16.405°
151 10376.571469275 C_{2} 0.000153836 0 0 12 139 0 0 447 298 0 16.163°
152 10517.867592878 D_{2} 0 0 0 12 140 0 0 450 300 0 16.117°
153 10660.082748237 D_{3} 0 0 0 12 141 0 0 453 302 0 16.390°
154 10803.372421141 C_{2} 0.000735800 0 0 12 142 0 0 456 304 0 16.078°
155 10947.574692279 C_{2} 0.000603670 0 0 12 143 0 0 459 306 0 15.990°
156 11092.798311456 C_{2} 0.000508534 0 0 12 144 0 0 462 308 0 15.822°
157 11238.903041156 C_{2} 0.000357679 0 0 12 145 0 0 465 310 0 15.948°
158 11385.990186197 C_{2} 0.000921918 0 0 12 146 0 0 468 312 0 15.987°
159 11534.023960956 C_{2} 0.000381457 0 0 12 147 0 0 471 314 0 15.960°
160 11683.054805549 D_{2} 0 0 0 12 148 0 0 474 316 0 15.961°
161 11833.084739465 C_{2} 0.000056447 0 0 12 149 0 0 477 318 0 15.810°
162 11984.050335814 D_{3} 0 0 0 12 150 0 0 480 320 0 15.813°
163 12136.013053220 C_{2} 0.000120798 0 0 12 151 0 0 483 322 0 15.675°
164 12288.930105320 D_{2} 0 0 0 12 152 0 0 486 324 0 15.655°
165 12442.804451373 C_{2} 0.000091119 0 0 12 153 0 0 489 326 0 15.651°
166 12597.649071323 D_{2d} 0 0 0 16 146 4 0 492 328 0 15.607°
167 12753.469429750 C_{2} 0.000097382 0 0 12 155 0 0 495 330 0 15.600°
168 12910.212672268 D_{3} 0 0 0 12 156 0 0 498 332 0 15.655°
169 13068.006451127 C_{s} 0.000068102 0 0 13 155 1 0 501 334 0 15.537°
170 13226.681078541 D_{2d} 0 0 0 12 158 0 0 504 336 0 15.569°
171 13386.355930717 D_{3} 0 0 0 12 159 0 0 507 338 0 15.497°
172 13547.018108787 C_{2v} 0.000547291 0 0 14 156 2 0 510 340 0 15.292°
173 13708.635243034 C_{s} 0.000286544 0 0 12 161 0 0 513 342 0 15.225°
174 13871.187092292 D_{2} 0 0 0 12 162 0 0 516 344 0 15.366°
175 14034.781306929 C_{2} 0.000026686 0 0 12 163 0 0 519 346 0 15.252°
176 14199.354775632 C_{1} 0.000283978 0 0 12 164 0 0 522 348 0 15.101°
177 14364.837545298 D_{5} 0 0 0 12 165 0 0 525 350 0 15.269°
178 14531.309552587 D_{2} 0 0 0 12 166 0 0 528 352 0 15.145°
179 14698.754594220 C_{1} 0.000125113 0 0 13 165 1 0 531 354 0 14.968°
180 14867.099927525 D_{2} 0 0 0 12 168 0 0 534 356 0 15.067°
181 15036.467239769 C_{2} 0.000304193 0 0 12 169 0 0 537 358 0 15.002°
182 15206.730610906 D_{5} 0 0 0 12 170 0 0 540 360 0 15.155°
183 15378.166571028 C_{1} 0.000467899 0 0 12 171 0 0 543 362 0 14.747°
184 15550.421450311 T 0 0 0 12 172 0 0 546 364 0 14.932°
185 15723.720074072 C_{2} 0.000389762 0 0 12 173 0 0 549 366 0 14.775°
186 15897.897437048 C_{1} 0.000389762 0 0 12 174 0 0 552 368 0 14.739°
187 16072.975186320 D_{5} 0 0 0 12 175 0 0 555 370 0 14.848°
188 16249.222678879 D_{2} 0 0 0 12 176 0 0 558 372 0 14.740°
189 16426.371938862 C_{2} 0.000020732 0 0 12 177 0 0 561 374 0 14.671°
190 16604.428338501 C_{3} 0.000586804 0 0 12 178 0 0 564 376 0 14.501°
191 16783.452219362 C_{1} 0.001129202 0 0 13 177 1 0 567 378 0 14.195°
192 16963.338386460 I 0 0 0 12 180 0 0 570 380 0 14.819°
193 17144.564740880 C_{2} 0.000985192 0 0 12 181 0 0 573 382 0 14.144°
194 17326.616136471 C_{1} 0.000322358 0 0 12 182 0 0 576 384 0 14.350°
195 17509.489303930 D_{3} 0 0 0 12 183 0 0 579 386 0 14.375°
196 17693.460548082 C_{2} 0.000315907 0 0 12 184 0 0 582 388 0 14.251°
197 17878.340162571 D_{5} 0 0 0 12 185 0 0 585 390 0 14.147°
198 18064.262177195 C_{2} 0.000011149 0 0 12 186 0 0 588 392 0 14.237°
199 18251.082495640 C_{1} 0.000534779 0 0 12 187 0 0 591 394 0 14.153°
200 18438.842717530 D_{2} 0 0 0 12 188 0 0 594 396 0 14.222°
201 18627.591226244 C_{1} 0.001048859 0 0 13 187 1 0 597 398 0 13.830°
202 18817.204718262 D_{5} 0 0 0 12 190 0 0 600 400 0 14.189°
203 19007.981204580 C_{s} 0.000600343 0 0 12 191 0 0 603 402 0 13.977°
204 19199.540775603 T_{h} 0 0 0 12 192 0 0 606 404 0 14.291°
212 20768.053085964 I 0 0 0 12 200 0 0 630 420 0 14.118°
214 21169.910410375 T 0 0 0 12 202 0 0 636 424 0 13.771°
216 21575.596377869 D_{3} 0 0 0 12 204 0 0 642 428 0 13.735°
217 21779.856080418 D_{5} 0 0 0 12 205 0 0 645 430 0 13.902°
232 24961.252318934 T 0 0 0 12 220 0 0 690 460 0 13.260°
255 30264.424251281 D_{3} 0 0 0 12 243 0 0 759 506 0 12.565°
256 30506.687515847 T 0 0 0 12 244 0 0 762 508 0 12.572°
257 30749.941417346 D_{5} 0 0 0 12 245 0 0 765 510 0 12.672°
272 34515.193292681 I_{h} 0 0 0 12 260 0 0 810 540 0 12.335°
282 37147.294418462 I 0 0 0 12 270 0 0 840 560 0 12.166°
292 39877.008012909 D_{5} 0 0 0 12 280 0 0 870 580 0 11.857°
306 43862.569780797 T_{h} 0 0 0 12 294 0 0 912 608 0 11.628°
312 45629.313804002 C_{2} 0.000306163 0 0 12 300 0 0 930 620 0 11.299°
315 46525.825643432 D_{3} 0 0 0 12 303 0 0 939 626 0 11.337°
317 47128.310344520 D_{5} 0 0 0 12 305 0 0 945 630 0 11.423°
318 47431.056020043 D_{3} 0 0 0 12 306 0 0 948 632 0 11.219°
334 52407.728127822 T 0 0 0 12 322 0 0 996 664 0 11.058°
348 56967.472454334 T_{h} 0 0 0 12 336 0 0 1038 692 0 10.721°
357 59999.922939598 D_{5} 0 0 0 12 345 0 0 1065 710 0 10.728°
358 60341.830924588 T 0 0 0 12 346 0 0 1068 712 0 10.647°
372 65230.027122557 I 0 0 0 12 360 0 0 1110 740 0 10.531°
382 68839.426839215 D_{5} 0 0 0 12 370 0 0 1140 760 0 10.379°
390 71797.035335953 T_{h} 0 0 0 12 378 0 0 1164 776 0 10.222°
392 72546.258370889 I 0 0 0 12 380 0 0 1170 780 0 10.278°
400 75582.448512213 T 0 0 0 12 388 0 0 1194 796 0 10.068°
402 76351.192432673 D_{5} 0 0 0 12 390 0 0 1200 800 0 10.099°
432 88353.709681956 D_{3} 0 0 0 24 396 12 0 1290 860 0 9.556°
448 95115.546986209 T 0 0 0 24 412 12 0 1338 892 0 9.322°
460 100351.763108673 T 0 0 0 24 424 12 0 1374 916 0 9.297°
468 103920.871715127 S_{6} 0 0 0 24 432 12 0 1398 932 0 9.120°
470 104822.886324279 S_{6} 0 0 0 24 434 12 0 1404 936 0 9.059°

References

  1. ↑ J. J. Thomson, "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure", Philosophical Magazine Series 6, Volume 7, Number 39, pp. 237–265, March 1904. [1]
  2. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  3. ↑ L. Foppl, "Stabile anordnungen von elektronen im atom", J. Reine Angew. Math, 141 (1912), 251–301.
  4. ↑ http://arxiv.org/abs/1001.3702
  5. ↑ V.A. Yudin, "The minimum of potential energy of a system of point charges", Discretnaya Matematika 4(2) (1992), 115–121 (in Russian); Discrete Math. Appl., 3(1) (1993), 75–81
  6. ↑ N.N. Andreev, "An extremal property of the icosahedron", East J. Approximation, 2(4) (1996), 459-462, MR 97m:52022, Zbl 0877.51021
  7. ↑ Y. Levin and J. J. Arenzon, ``Why charges go to the Surface: A generalized Thomson Problem Europhys. Lett. Vol. 63 p. 415 (2003)
  8. ↑ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
  9. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  10. ↑ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.

Notes

  • Henry Cohn and Abhinav Kumar, "Universally optimal distribution of points on spheres". J. Amer. Math. Soc. 20 (2007), no. 1, 99—148
  • P. D. Dragnev, D. A. Legg, and D. W. Townsend, "Discrete logarithmic energy on the sphere". Pacific J. Math. 207 (2002), no. 2, 345—358.
  • T. Erber and G. M. Hockney, "Complex Systems: Equilibrium Configurations of N Equal Charges on a Sphere (2\leq N\leq 112)", Advances in Chemical Physics, Volume 98, pp. 495–594, 1997.
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