# Thomson problem

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.

## 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.

## 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]

 "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.

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
2. ^ S. Smale (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer 20 (2): 7–15.
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. ^ LaFave Jr, Tim (December 2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure". Journal of Electrostatics 71 (6): 1029–1035. doi:10.1016/j.elstat.2013.10.001. Retrieved 10-Feb-2014.

## 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.