# Thomson problem

The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of n electrons constrained to 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 electrostatic interaction energy occurring between each pair of electrons of equal charges (${\displaystyle e_{i}=e_{j}=e}$, with ${\displaystyle e}$ the elementary charge of an electron) is given by Coulomb's Law,

${\displaystyle U_{ij}(N)=k_{\text{e}}{e_{i}e_{j} \over r_{ij}}.}$

Here, ${\displaystyle k_{\text{e}}}$ is the Coulomb constant and ${\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}$ is the distance between each pair of electrons located at points on the sphere defined by vectors ${\displaystyle \mathbf {r} _{i}}$ and ${\displaystyle \mathbf {r} _{j}}$, respectively.

Simplified units of ${\displaystyle e=1}$ and ${\displaystyle k_{e}=1}$ are used without loss of generality. Then,

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

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

${\displaystyle U(N)=\sum _{i

The global minimization of ${\displaystyle U(N)}$ over all possible configurations of N distinct points is typically found by numerical minimization algorithms.

Thomson's problem is related to the 7th of the eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere".[2] The main difference is that in Smale's problem the function to minimise is not the electrostatic potential ${\displaystyle 1 \over r_{ij}}$ but a logarithmic potential given by ${\displaystyle -\log r_{ij}.}$ A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number N of points goes to infinity, not for concrete values of N.

### 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, ${\displaystyle r_{ij}=2r=2}$, or

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

## Known exact solutions

Schematic geometric solutions of the mathematical Thomson Problem for up to N = 5 electrons.

Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases.

• For N = 1, the solution is trivial. The single electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero because the electron experiences no electric field due to other sources of charge.
• For N = 2, the optimal configuration consists of electrons at antipodal points. This represents the first one-dimensional solution.
• For N = 3, electrons reside at the vertices of an equilateral triangle about any great circle.[3] The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. Also, this represents the first two-dimensional solution.
• For N = 4, electrons reside at the vertices of a regular tetrahedron. Of interest, this represents the first three-dimensional solution.
• For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid.[4] Of interest, it is impossible for any N solution with five or more electrons to exhibit global equidistance among all pairs of electrons.
• For N = 6, electrons reside at vertices of a regular octahedron.[5] The configuration may be imagined as four electrons residing at the corners of a square about the equator and the remaining two residing at the poles.
• For N = 12, electrons reside at the vertices of a regular icosahedron.[6]

Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are 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 and pentagonal, respectively.[citation needed]

## 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

${\displaystyle \sum _{i

Traditionally, one considers ${\displaystyle f(x)=x^{-\alpha }}$ also known as Riesz ${\displaystyle \alpha }$-kernels. For integrable Riesz kernels see the 1972 work of Landkof.[7] For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.[8] Notable cases include:[9]

• α = ∞, the Tammes problem (packing);
• α = 1, the Thomson problem;
• α = 0, to maximize the product of distances, latterly known as Whyte's problem;
• α = −1 : maximum average distance problem.

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

## Solution algorithms

Several algorithms have been applied to this problem. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance:[9]

• constrained global optimization (Altschuler et al. 1994),
• steepest descent (Claxton and Benson 1966, Erber and Hockney 1991),
• random walk (Weinrach et al. 1990),
• genetic algorithm (Morris et al. 1996)

While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest.

### Continuous spherical shell charge

The extreme upper energy limit of the Thomson Problem is given by ${\displaystyle N^{2}/2}$ for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Significantly lower energy of a given N-electron solution of the Thomson Problem with one charge at its origin is readily obtained by ${\displaystyle U(N)+N}$, where ${\displaystyle U(N)}$ are solutions of the Thomson Problem.

The energy of a continuous spherical shell of charge distributed across its surface is given by

${\displaystyle U_{\text{shell}}(N)={\frac {N^{2}}{2}}}$

and is, in general, greater than the energy of every Thomson problem solution. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. For example, a spherical shell of ${\displaystyle N=1}$ represents the uniform distribution of a single electron's charge, ${\displaystyle -e}$ across the entire shell.

### Randomly distributed point charges

The global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by

${\displaystyle U_{\text{rand}}(N)={\frac {N(N-1)}{2}}}$

and is, in general, greater than the energy of every Thomson problem solution.

Here, N is a discrete variable that counts the number of electrons in the system. As well, ${\displaystyle U_{\text{rand}}(N)

### Charge-centered distribution

For every Nth solution of the Thomson problem there is an ${\displaystyle (N+1)}$th configuration that includes an electron at the origin of the sphere whose energy is simply the addition of N to the energy of the Nth solution. That is,[10]

${\displaystyle U_{0}(N+1)=U_{\text{Thom}}(N)+N.}$

Thus, if ${\displaystyle U_{\text{Thom}}(N)}$ is known exactly, then ${\displaystyle U_{0}(N+1)}$ is known exactly.

In general, ${\displaystyle U_{0}(N+1)}$ is greater than ${\displaystyle U_{\text{Thom}}(N+1)}$, but is remarkably closer to each ${\displaystyle (N+1)}$th Thomson solution than ${\displaystyle U_{\text{shell}}(N+1)}$ and ${\displaystyle U_{\text{rand}}(N+1)}$. Therefore, the charge-centered distribution represents a smaller "energy gap" to cross to arrive at a solution of each Thomson problem than algorithms that begin with the other two charge configurations.

## Relations to other scientific problems

The Thomson problem is a natural consequence of J. J. Thomson's plum pudding model in the absence of its uniform positive background charge.[11]

"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[12]

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.[13]

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 arrangements of protein subunits that 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 Abrikosov vortices that form at low temperatures in a superconducting metal shell with a large monopole at its center.

## Configurations of smallest known energy

In the following table ${\displaystyle N}$ is the number of points (charges) in a configuration, ${\displaystyle U_{\textrm {Thom}}}$ is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and ${\displaystyle r_{i}}$ 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, ${\displaystyle v_{i}}$ is the number of vertices where the given number of edges meet, ${\displaystyle e}$ is the total number of edges, ${\displaystyle f_{3}}$ is the number of triangular faces, ${\displaystyle f_{4}}$ is the number of quadrilateral faces, and ${\displaystyle \theta _{1}}$ 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 = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last column.[14]

N ${\displaystyle U_{\textrm {Thom}}}$ Symmetry ${\displaystyle \left|\sum \mathbf {r} _{i}\right|}$ ${\displaystyle v_{3}}$ ${\displaystyle v_{4}}$ ${\displaystyle v_{5}}$ ${\displaystyle v_{6}}$ ${\displaystyle v_{7}}$ ${\displaystyle v_{8}}$ ${\displaystyle e}$ ${\displaystyle f_{3}}$ ${\displaystyle f_{4}}$ ${\displaystyle \theta _{1}}$ Equivalent polyhedron
2 0.500000000 ${\displaystyle D_{\infty h}}$ 0 2 180.000° digon
3 1.732050808 ${\displaystyle D_{3h}}$ 0 3 2 120.000° triangle
4 3.674234614 ${\displaystyle T_{d}}$ 0 4 0 0 0 0 0 6 4 0 109.471° tetrahedron
5 6.474691495 ${\displaystyle D_{3h}}$ 0 2 3 0 0 0 0 9 6 0 90.000° triangular dipyramid
6 9.985281374 ${\displaystyle O_{h}}$ 0 0 6 0 0 0 0 12 8 0 90.000° octahedron
7 14.452977414 ${\displaystyle D_{5h}}$ 0 0 5 2 0 0 0 15 10 0 72.000° pentagonal dipyramid
8 19.675287861 ${\displaystyle D_{4d}}$ 0 0 8 0 0 0 0 16 8 2 71.694° square antiprism
9 25.759986531 ${\displaystyle D_{3h}}$ 0 0 3 6 0 0 0 21 14 0 69.190° triaugmented triangular prism
10 32.716949460 ${\displaystyle D_{4d}}$ 0 0 2 8 0 0 0 24 16 0 64.996° gyroelongated square dipyramid
11 40.596450510 ${\displaystyle C_{2v}}$ 0.013219635 0 2 8 1 0 0 27 18 0 58.540° edge-contracted icosahedron
12 49.165253058 ${\displaystyle I_{h}}$ 0 0 0 12 0 0 0 30 20 0 63.435° icosahedron
(geodesic sphere {3,5+}1,0)
13 58.853230612 ${\displaystyle C_{2v}}$ 0.008820367 0 1 10 2 0 0 33 22 0 52.317°
14 69.306363297 ${\displaystyle D_{6d}}$ 0 0 0 12 2 0 0 36 24 0 52.866° gyroelongated hexagonal dipyramid
15 80.670244114 ${\displaystyle D_{3}}$ 0 0 0 12 3 0 0 39 26 0 49.225°
16 92.911655302 ${\displaystyle T}$ 0 0 0 12 4 0 0 42 28 0 48.936°
17 106.050404829 ${\displaystyle D_{5h}}$ 0 0 0 12 5 0 0 45 30 0 50.108° double-gyroelongated pentagonal dipyramid
18 120.084467447 ${\displaystyle D_{4d}}$ 0 0 2 8 8 0 0 48 32 0 47.534°
19 135.089467557 ${\displaystyle C_{2v}}$ 0.000135163 0 0 14 5 0 0 50 32 1 44.910°
20 150.881568334 ${\displaystyle D_{3h}}$ 0 0 0 12 8 0 0 54 36 0 46.093°
21 167.641622399 ${\displaystyle C_{2v}}$ 0.001406124 0 1 10 10 0 0 57 38 0 44.321°
22 185.287536149 ${\displaystyle T_{d}}$ 0 0 0 12 10 0 0 60 40 0 43.302°
23 203.930190663 ${\displaystyle D_{3}}$ 0 0 0 12 11 0 0 63 42 0 41.481°
24 223.347074052 ${\displaystyle O}$ 0 0 0 24 0 0 0 60 32 6 42.065° snub cube
25 243.812760299 ${\displaystyle C_{s}}$ 0.001021305 0 0 14 11 0 0 68 44 1 39.610°
26 265.133326317 ${\displaystyle C_{2}}$ 0.001919065 0 0 12 14 0 0 72 48 0 38.842°
27 287.302615033 ${\displaystyle D_{5h}}$ 0 0 0 12 15 0 0 75 50 0 39.940°
28 310.491542358 ${\displaystyle T}$ 0 0 0 12 16 0 0 78 52 0 37.824°
29 334.634439920 ${\displaystyle D_{3}}$ 0 0 0 12 17 0 0 81 54 0 36.391°
30 359.603945904 ${\displaystyle D_{2}}$ 0 0 0 12 18 0 0 84 56 0 36.942°
31 385.530838063 ${\displaystyle C_{3v}}$ 0.003204712 0 0 12 19 0 0 87 58 0 36.373°
32 412.261274651 ${\displaystyle I_{h}}$ 0 0 0 12 20 0 0 90 60 0 37.377° pentakis dodecahedron
(geodesic sphere {3,5+}1,1)
33 440.204057448 ${\displaystyle C_{s}}$ 0.004356481 0 0 15 17 1 0 92 60 1 33.700°
34 468.904853281 ${\displaystyle D_{2}}$ 0 0 0 12 22 0 0 96 64 0 33.273°
35 498.569872491 ${\displaystyle C_{2}}$ 0.000419208 0 0 12 23 0 0 99 66 0 33.100°
36 529.122408375 ${\displaystyle D_{2}}$ 0 0 0 12 24 0 0 102 68 0 33.229°
37 560.618887731 ${\displaystyle D_{5h}}$ 0 0 0 12 25 0 0 105 70 0 32.332°
38 593.038503566 ${\displaystyle D_{6d}}$ 0 0 0 12 26 0 0 108 72 0 33.236°
39 626.389009017 ${\displaystyle D_{3h}}$ 0 0 0 12 27 0 0 111 74 0 32.053°
40 660.675278835 ${\displaystyle T_{d}}$ 0 0 0 12 28 0 0 114 76 0 31.916°
41 695.916744342 ${\displaystyle D_{3h}}$ 0 0 0 12 29 0 0 117 78 0 31.528°
42 732.078107544 ${\displaystyle D_{5h}}$ 0 0 0 12 30 0 0 120 80 0 31.245°
43 769.190846459 ${\displaystyle C_{2v}}$ 0.000399668 0 0 12 31 0 0 123 82 0 30.867°
44 807.174263085 ${\displaystyle O_{h}}$ 0 0 0 24 20 0 0 120 72 6 31.258°
45 846.188401061 ${\displaystyle D_{3}}$ 0 0 0 12 33 0 0 129 86 0 30.207°
46 886.167113639 ${\displaystyle T}$ 0 0 0 12 34 0 0 132 88 0 29.790°
47 927.059270680 ${\displaystyle C_{s}}$ 0.002482914 0 0 14 33 0 0 134 88 1 28.787°
48 968.713455344 ${\displaystyle O}$ 0 0 0 24 24 0 0 132 80 6 29.690°
49 1011.557182654 ${\displaystyle C_{3}}$ 0.001529341 0 0 12 37 0 0 141 94 0 28.387°
50 1055.182314726 ${\displaystyle D_{6d}}$ 0 0 0 12 38 0 0 144 96 0 29.231°
51 1099.819290319 ${\displaystyle D_{3}}$ 0 0 0 12 39 0 0 147 98 0 28.165°
52 1145.418964319 ${\displaystyle C_{3}}$ 0.000457327 0 0 12 40 0 0 150 100 0 27.670°
53 1191.922290416 ${\displaystyle C_{2v}}$ 0.000278469 0 0 18 35 0 0 150 96 3 27.137°
54 1239.361474729 ${\displaystyle C_{2}}$ 0.000137870 0 0 12 42 0 0 156 104 0 27.030°
55 1287.772720783 ${\displaystyle C_{2}}$ 0.000391696 0 0 12 43 0 0 159 106 0 26.615°
56 1337.094945276 ${\displaystyle D_{2}}$ 0 0 0 12 44 0 0 162 108 0 26.683°
57 1387.383229253 ${\displaystyle D_{3}}$ 0 0 0 12 45 0 0 165 110 0 26.702°
58 1438.618250640 ${\displaystyle D_{2}}$ 0 0 0 12 46 0 0 168 112 0 26.155°
59 1490.773335279 ${\displaystyle C_{2}}$ 0.000154286 0 0 14 43 2 0 171 114 0 26.170°
60 1543.830400976 ${\displaystyle D_{3}}$ 0 0 0 12 48 0 0 174 116 0 25.958°
61 1597.941830199 ${\displaystyle C_{1}}$ 0.001091717 0 0 12 49 0 0 177 118 0 25.392°
62 1652.909409898 ${\displaystyle D_{5}}$ 0 0 0 12 50 0 0 180 120 0 25.880°
63 1708.879681503 ${\displaystyle D_{3}}$ 0 0 0 12 51 0 0 183 122 0 25.257°
64 1765.802577927 ${\displaystyle D_{2}}$ 0 0 0 12 52 0 0 186 124 0 24.920°
65 1823.667960264 ${\displaystyle C_{2}}$ 0.000399515 0 0 12 53 0 0 189 126 0 24.527°
66 1882.441525304 ${\displaystyle C_{2}}$ 0.000776245 0 0 12 54 0 0 192 128 0 24.765°
67 1942.122700406 ${\displaystyle D_{5}}$ 0 0 0 12 55 0 0 195 130 0 24.727°
68 2002.874701749 ${\displaystyle D_{2}}$ 0 0 0 12 56 0 0 198 132 0 24.433°
69 2064.533483235 ${\displaystyle D_{3}}$ 0 0 0 12 57 0 0 201 134 0 24.137°
70 2127.100901551 ${\displaystyle D_{2d}}$ 0 0 0 12 50 0 0 200 128 4 24.291°
71 2190.649906425 ${\displaystyle C_{2}}$ 0.001256769 0 0 14 55 2 0 207 138 0 23.803°
72 2255.001190975 ${\displaystyle I}$ 0 0 0 12 60 0 0 210 140 0 24.492° geodesic sphere {3,5+}2,1
73 2320.633883745 ${\displaystyle C_{2}}$ 0.001572959 0 0 12 61 0 0 213 142 0 22.810°
74 2387.072981838 ${\displaystyle C_{2}}$ 0.000641539 0 0 12 62 0 0 216 144 0 22.966°
75 2454.369689040 ${\displaystyle D_{3}}$ 0 0 0 12 63 0 0 219 146 0 22.736°
76 2522.674871841 ${\displaystyle C_{2}}$ 0.000943474 0 0 12 64 0 0 222 148 0 22.886°
77 2591.850152354 ${\displaystyle D_{5}}$ 0 0 0 12 65 0 0 225 150 0 23.286°
78 2662.046474566 ${\displaystyle T_{h}}$ 0 0 0 12 66 0 0 228 152 0 23.426°
79 2733.248357479 ${\displaystyle C_{s}}$ 0.000702921 0 0 12 63 1 0 230 152 1 22.636°
80 2805.355875981 ${\displaystyle D_{4d}}$ 0 0 0 16 64 0 0 232 152 2 22.778°
81 2878.522829664 ${\displaystyle C_{2}}$ 0.000194289 0 0 12 69 0 0 237 158 0 21.892°
82 2952.569675286 ${\displaystyle D_{2}}$ 0 0 0 12 70 0 0 240 160 0 22.206°
83 3027.528488921 ${\displaystyle C_{2}}$ 0.000339815 0 0 14 67 2 0 243 162 0 21.646°
84 3103.465124431 ${\displaystyle C_{2}}$ 0.000401973 0 0 12 72 0 0 246 164 0 21.513°
85 3180.361442939 ${\displaystyle C_{2}}$ 0.000416581 0 0 12 73 0 0 249 166 0 21.498°
86 3258.211605713 ${\displaystyle C_{2}}$ 0.001378932 0 0 12 74 0 0 252 168 0 21.522°
87 3337.000750014 ${\displaystyle C_{2}}$ 0.000754863 0 0 12 75 0 0 255 170 0 21.456°
88 3416.720196758 ${\displaystyle D_{2}}$ 0 0 0 12 76 0 0 258 172 0 21.486°
89 3497.439018625 ${\displaystyle C_{2}}$ 0.000070891 0 0 12 77 0 0 261 174 0 21.182°
90 3579.091222723 ${\displaystyle D_{3}}$ 0 0 0 12 78 0 0 264 176 0 21.230°
91 3661.713699320 ${\displaystyle C_{2}}$ 0.000033221 0 0 12 79 0 0 267 178 0 21.105°
92 3745.291636241 ${\displaystyle D_{2}}$ 0 0 0 12 80 0 0 270 180 0 21.026°
93 3829.844338421 ${\displaystyle C_{2}}$ 0.000213246 0 0 12 81 0 0 273 182 0 20.751°
94 3915.309269620 ${\displaystyle D_{2}}$ 0 0 0 12 82 0 0 276 184 0 20.952°
95 4001.771675565 ${\displaystyle C_{2}}$ 0.000116638 0 0 12 83 0 0 279 186 0 20.711°
96 4089.154010060 ${\displaystyle C_{2}}$ 0.000036310 0 0 12 84 0 0 282 188 0 20.687°
97 4177.533599622 ${\displaystyle C_{2}}$ 0.000096437 0 0 12 85 0 0 285 190 0 20.450°
98 4266.822464156 ${\displaystyle C_{2}}$ 0.000112916 0 0 12 86 0 0 288 192 0 20.422°
99 4357.139163132 ${\displaystyle C_{2}}$ 0.000156508 0 0 12 87 0 0 291 194 0 20.284°
100 4448.350634331 ${\displaystyle T}$ 0 0 0 12 88 0 0 294 196 0 20.297°
101 4540.590051694 ${\displaystyle D_{3}}$ 0 0 0 12 89 0 0 297 198 0 20.011°
102 4633.736565899 ${\displaystyle D_{3}}$ 0 0 0 12 90 0 0 300 200 0 20.040°
103 4727.836616833 ${\displaystyle C_{2}}$ 0.000201245 0 0 12 91 0 0 303 202 0 19.907°
104 4822.876522746 ${\displaystyle D_{6}}$ 0 0 0 12 92 0 0 306 204 0 19.957°
105 4919.000637616 ${\displaystyle D_{3}}$ 0 0 0 12 93 0 0 309 206 0 19.842°
106 5015.984595705 ${\displaystyle D_{2}}$ 0 0 0 12 94 0 0 312 208 0 19.658°
107 5113.953547724 ${\displaystyle C_{2}}$ 0.000064137 0 0 12 95 0 0 315 210 0 19.327°
108 5212.813507831 ${\displaystyle C_{2}}$ 0.000432525 0 0 12 96 0 0 318 212 0 19.327°
109 5312.735079920 ${\displaystyle C_{2}}$ 0.000647299 0 0 14 93 2 0 321 214 0 19.103°
110 5413.549294192 ${\displaystyle D_{6}}$ 0 0 0 12 98 0 0 324 216 0 19.476°
111 5515.293214587 ${\displaystyle D_{3}}$ 0 0 0 12 99 0 0 327 218 0 19.255°
112 5618.044882327 ${\displaystyle D_{5}}$ 0 0 0 12 100 0 0 330 220 0 19.351°
113 5721.824978027 ${\displaystyle D_{3}}$ 0 0 0 12 101 0 0 333 222 0 18.978°
114 5826.521572163 ${\displaystyle C_{2}}$ 0.000149772 0 0 12 102 0 0 336 224 0 18.836°
115 5932.181285777 ${\displaystyle C_{3}}$ 0.000049972 0 0 12 103 0 0 339 226 0 18.458°
116 6038.815593579 ${\displaystyle C_{2}}$ 0.000259726 0 0 12 104 0 0 342 228 0 18.386°
117 6146.342446579 ${\displaystyle C_{2}}$ 0.000127609 0 0 12 105 0 0 345 230 0 18.566°
118 6254.877027790 ${\displaystyle C_{2}}$ 0.000332475 0 0 12 106 0 0 348 232 0 18.455°
119 6364.347317479 ${\displaystyle C_{2}}$ 0.000685590 0 0 12 107 0 0 351 234 0 18.336°
120 6474.756324980 ${\displaystyle C_{s}}$ 0.001373062 0 0 12 108 0 0 354 236 0 18.418°
121 6586.121949584 ${\displaystyle C_{3}}$ 0.000838863 0 0 12 109 0 0 357 238 0 18.199°
122 6698.374499261 ${\displaystyle I_{h}}$ 0 0 0 12 110 0 0 360 240 0 18.612° geodesic sphere {3,5+}2,2
123 6811.827228174 ${\displaystyle C_{2v}}$ 0.001939754 0 0 14 107 2 0 363 242 0 17.840°
124 6926.169974193 ${\displaystyle D_{2}}$ 0 0 0 12 112 0 0 366 244 0 18.111°
125 7041.473264023 ${\displaystyle C_{2}}$ 0.000088274 0 0 12 113 0 0 369 246 0 17.867°
126 7157.669224867 ${\displaystyle D_{4}}$ 0 0 2 16 100 8 0 372 248 0 17.920°
127 7274.819504675 ${\displaystyle D_{5}}$ 0 0 0 12 115 0 0 375 250 0 17.877°
128 7393.007443068 ${\displaystyle C_{2}}$ 0.000054132 0 0 12 116 0 0 378 252 0 17.814°
129 7512.107319268 ${\displaystyle C_{2}}$ 0.000030099 0 0 12 117 0 0 381 254 0 17.743°
130 7632.167378912 ${\displaystyle C_{2}}$ 0.000025622 0 0 12 118 0 0 384 256 0 17.683°
131 7753.205166941 ${\displaystyle C_{2}}$ 0.000305133 0 0 12 119 0 0 387 258 0 17.511°
132 7875.045342797 ${\displaystyle I}$ 0 0 0 12 120 0 0 390 260 0 17.958° geodesic sphere {3,5+}3,1
133 7998.179212898 ${\displaystyle C_{3}}$ 0.000591438 0 0 12 121 0 0 393 262 0 17.133°
134 8122.089721194 ${\displaystyle C_{2}}$ 0.000470268 0 0 12 122 0 0 396 264 0 17.214°
135 8246.909486992 ${\displaystyle D_{3}}$ 0 0 0 12 123 0 0 399 266 0 17.431°
136 8372.743302539 ${\displaystyle T}$ 0 0 0 12 124 0 0 402 268 0 17.485°
137 8499.534494782 ${\displaystyle D_{5}}$ 0 0 0 12 125 0 0 405 270 0 17.560°
138 8627.406389880 ${\displaystyle C_{2}}$ 0.000473576 0 0 12 126 0 0 408 272 0 16.924°
139 8756.227056057 ${\displaystyle C_{2}}$ 0.000404228 0 0 12 127 0 0 411 274 0 16.673°
140 8885.980609041 ${\displaystyle C_{1}}$ 0.000630351 0 0 13 126 1 0 414 276 0 16.773°
141 9016.615349190 ${\displaystyle C_{2v}}$ 0.000376365 0 0 14 126 0 1 417 278 0 16.962°
142 9148.271579993 ${\displaystyle C_{2}}$ 0.000550138 0 0 12 130 0 0 420 280 0 16.840°
143 9280.839851192 ${\displaystyle C_{2}}$ 0.000255449 0 0 12 131 0 0 423 282 0 16.782°
144 9414.371794460 ${\displaystyle D_{2}}$ 0 0 0 12 132 0 0 426 284 0 16.953°
145 9548.928837232 ${\displaystyle C_{s}}$ 0.000094938 0 0 12 133 0 0 429 286 0 16.841°
146 9684.381825575 ${\displaystyle D_{2}}$ 0 0 0 12 134 0 0 432 288 0 16.905°
147 9820.932378373 ${\displaystyle C_{2}}$ 0.000636651 0 0 12 135 0 0 435 290 0 16.458°
148 9958.406004270 ${\displaystyle C_{2}}$ 0.000203701 0 0 12 136 0 0 438 292 0 16.627°
149 10096.859907397 ${\displaystyle C_{1}}$ 0.000638186 0 0 14 133 2 0 441 294 0 16.344°
150 10236.196436701 ${\displaystyle T}$ 0 0 0 12 138 0 0 444 296 0 16.405°
151 10376.571469275 ${\displaystyle C_{2}}$ 0.000153836 0 0 12 139 0 0 447 298 0 16.163°
152 10517.867592878 ${\displaystyle D_{2}}$ 0 0 0 12 140 0 0 450 300 0 16.117°
153 10660.082748237 ${\displaystyle D_{3}}$ 0 0 0 12 141 0 0 453 302 0 16.390°
154 10803.372421141 ${\displaystyle C_{2}}$ 0.000735800 0 0 12 142 0 0 456 304 0 16.078°
155 10947.574692279 ${\displaystyle C_{2}}$ 0.000603670 0 0 12 143 0 0 459 306 0 15.990°
156 11092.798311456 ${\displaystyle C_{2}}$ 0.000508534 0 0 12 144 0 0 462 308 0 15.822°
157 11238.903041156 ${\displaystyle C_{2}}$ 0.000357679 0 0 12 145 0 0 465 310 0 15.948°
158 11385.990186197 ${\displaystyle C_{2}}$ 0.000921918 0 0 12 146 0 0 468 312 0 15.987°
159 11534.023960956 ${\displaystyle C_{2}}$ 0.000381457 0 0 12 147 0 0 471 314 0 15.960°
160 11683.054805549 ${\displaystyle D_{2}}$ 0 0 0 12 148 0 0 474 316 0 15.961°
161 11833.084739465 ${\displaystyle C_{2}}$ 0.000056447 0 0 12 149 0 0 477 318 0 15.810°
162 11984.050335814 ${\displaystyle D_{3}}$ 0 0 0 12 150 0 0 480 320 0 15.813°
163 12136.013053220 ${\displaystyle C_{2}}$ 0.000120798 0 0 12 151 0 0 483 322 0 15.675°
164 12288.930105320 ${\displaystyle D_{2}}$ 0 0 0 12 152 0 0 486 324 0 15.655°
165 12442.804451373 ${\displaystyle C_{2}}$ 0.000091119 0 0 12 153 0 0 489 326 0 15.651°
166 12597.649071323 ${\displaystyle D_{2d}}$ 0 0 0 16 146 4 0 492 328 0 15.607°
167 12753.469429750 ${\displaystyle C_{2}}$ 0.000097382 0 0 12 155 0 0 495 330 0 15.600°
168 12910.212672268 ${\displaystyle D_{3}}$ 0 0 0 12 156 0 0 498 332 0 15.655°
169 13068.006451127 ${\displaystyle C_{s}}$ 0.000068102 0 0 13 155 1 0 501 334 0 15.537°
170 13226.681078541 ${\displaystyle D_{2d}}$ 0 0 0 12 158 0 0 504 336 0 15.569°
171 13386.355930717 ${\displaystyle D_{3}}$ 0 0 0 12 159 0 0 507 338 0 15.497°
172 13547.018108787 ${\displaystyle C_{2v}}$ 0.000547291 0 0 14 156 2 0 510 340 0 15.292°
173 13708.635243034 ${\displaystyle C_{s}}$ 0.000286544 0 0 12 161 0 0 513 342 0 15.225°
174 13871.187092292 ${\displaystyle D_{2}}$ 0 0 0 12 162 0 0 516 344 0 15.366°
175 14034.781306929 ${\displaystyle C_{2}}$ 0.000026686 0 0 12 163 0 0 519 346 0 15.252°
176 14199.354775632 ${\displaystyle C_{1}}$ 0.000283978 0 0 12 164 0 0 522 348 0 15.101°
177 14364.837545298 ${\displaystyle D_{5}}$ 0 0 0 12 165 0 0 525 350 0 15.269°
178 14531.309552587 ${\displaystyle D_{2}}$ 0 0 0 12 166 0 0 528 352 0 15.145°
179 14698.754594220 ${\displaystyle C_{1}}$ 0.000125113 0 0 13 165 1 0 531 354 0 14.968°
180 14867.099927525 ${\displaystyle D_{2}}$ 0 0 0 12 168 0 0 534 356 0 15.067°
181 15036.467239769 ${\displaystyle C_{2}}$ 0.000304193 0 0 12 169 0 0 537 358 0 15.002°
182 15206.730610906 ${\displaystyle D_{5}}$ 0 0 0 12 170 0 0 540 360 0 15.155°
183 15378.166571028 ${\displaystyle C_{1}}$ 0.000467899 0 0 12 171 0 0 543 362 0 14.747°
184 15550.421450311 ${\displaystyle T}$ 0 0 0 12 172 0 0 546 364 0 14.932°
185 15723.720074072 ${\displaystyle C_{2}}$ 0.000389762 0 0 12 173 0 0 549 366 0 14.775°
186 15897.897437048 ${\displaystyle C_{1}}$ 0.000389762 0 0 12 174 0 0 552 368 0 14.739°
187 16072.975186320 ${\displaystyle D_{5}}$ 0 0 0 12 175 0 0 555 370 0 14.848°
188 16249.222678879 ${\displaystyle D_{2}}$ 0 0 0 12 176 0 0 558 372 0 14.740°
189 16426.371938862 ${\displaystyle C_{2}}$ 0.000020732 0 0 12 177 0 0 561 374 0 14.671°
190 16604.428338501 ${\displaystyle C_{3}}$ 0.000586804 0 0 12 178 0 0 564 376 0 14.501°
191 16783.452219362 ${\displaystyle C_{1}}$ 0.001129202 0 0 13 177 1 0 567 378 0 14.195°
192 16963.338386460 ${\displaystyle I}$ 0 0 0 12 180 0 0 570 380 0 14.819° geodesic sphere {3,5+}3,2
193 17144.564740880 ${\displaystyle C_{2}}$ 0.000985192 0 0 12 181 0 0 573 382 0 14.144°
194 17326.616136471 ${\displaystyle C_{1}}$ 0.000322358 0 0 12 182 0 0 576 384 0 14.350°
195 17509.489303930 ${\displaystyle D_{3}}$ 0 0 0 12 183 0 0 579 386 0 14.375°
196 17693.460548082 ${\displaystyle C_{2}}$ 0.000315907 0 0 12 184 0 0 582 388 0 14.251°
197 17878.340162571 ${\displaystyle D_{5}}$ 0 0 0 12 185 0 0 585 390 0 14.147°
198 18064.262177195 ${\displaystyle C_{2}}$ 0.000011149 0 0 12 186 0 0 588 392 0 14.237°
199 18251.082495640 ${\displaystyle C_{1}}$ 0.000534779 0 0 12 187 0 0 591 394 0 14.153°
200 18438.842717530 ${\displaystyle D_{2}}$ 0 0 0 12 188 0 0 594 396 0 14.222°
201 18627.591226244 ${\displaystyle C_{1}}$ 0.001048859 0 0 13 187 1 0 597 398 0 13.830°
202 18817.204718262 ${\displaystyle D_{5}}$ 0 0 0 12 190 0 0 600 400 0 14.189°
203 19007.981204580 ${\displaystyle C_{s}}$ 0.000600343 0 0 12 191 0 0 603 402 0 13.977°
204 19199.540775603 ${\displaystyle T_{h}}$ 0 0 0 12 192 0 0 606 404 0 14.291°
212 20768.053085964 ${\displaystyle I}$ 0 0 0 12 200 0 0 630 420 0 14.118° geodesic sphere {3,5+}4,1
214 21169.910410375 ${\displaystyle T}$ 0 0 0 12 202 0 0 636 424 0 13.771°
216 21575.596377869 ${\displaystyle D_{3}}$ 0 0 0 12 204 0 0 642 428 0 13.735°
217 21779.856080418 ${\displaystyle D_{5}}$ 0 0 0 12 205 0 0 645 430 0 13.902°
232 24961.252318934 ${\displaystyle T}$ 0 0 0 12 220 0 0 690 460 0 13.260°
255 30264.424251281 ${\displaystyle D_{3}}$ 0 0 0 12 243 0 0 759 506 0 12.565°
256 30506.687515847 ${\displaystyle T}$ 0 0 0 12 244 0 0 762 508 0 12.572°
257 30749.941417346 ${\displaystyle D_{5}}$ 0 0 0 12 245 0 0 765 510 0 12.672°
272 34515.193292681 ${\displaystyle I_{h}}$ 0 0 0 12 260 0 0 810 540 0 12.335° geodesic sphere {3,5+}3,3
282 37147.294418462 ${\displaystyle I}$ 0 0 0 12 270 0 0 840 560 0 12.166° geodesic sphere {3,5+}4,2
292 39877.008012909 ${\displaystyle D_{5}}$ 0 0 0 12 280 0 0 870 580 0 11.857°
306 43862.569780797 ${\displaystyle T_{h}}$ 0 0 0 12 294 0 0 912 608 0 11.628°
312 45629.313804002 ${\displaystyle C_{2}}$ 0.000306163 0 0 12 300 0 0 930 620 0 11.299°
315 46525.825643432 ${\displaystyle D_{3}}$ 0 0 0 12 303 0 0 939 626 0 11.337°
317 47128.310344520 ${\displaystyle D_{5}}$ 0 0 0 12 305 0 0 945 630 0 11.423°
318 47431.056020043 ${\displaystyle D_{3}}$ 0 0 0 12 306 0 0 948 632 0 11.219°
334 52407.728127822 ${\displaystyle T}$ 0 0 0 12 322 0 0 996 664 0 11.058°
348 56967.472454334 ${\displaystyle T_{h}}$ 0 0 0 12 336 0 0 1038 692 0 10.721°
357 59999.922939598 ${\displaystyle D_{5}}$ 0 0 0 12 345 0 0 1065 710 0 10.728°
358 60341.830924588 ${\displaystyle T}$ 0 0 0 12 346 0 0 1068 712 0 10.647°
372 65230.027122557 ${\displaystyle I}$ 0 0 0 12 360 0 0 1110 740 0 10.531° geodesic sphere {3,5+}4,3
382 68839.426839215 ${\displaystyle D_{5}}$ 0 0 0 12 370 0 0 1140 760 0 10.379°
390 71797.035335953 ${\displaystyle T_{h}}$ 0 0 0 12 378 0 0 1164 776 0 10.222°
392 72546.258370889 ${\displaystyle C_{1}}$ 0 0 0 12 380 0 0 1170 780 0 10.278°
400 75582.448512213 ${\displaystyle T}$ 0 0 0 12 388 0 0 1194 796 0 10.068°
402 76351.192432673 ${\displaystyle D_{5}}$ 0 0 0 12 390 0 0 1200 800 0 10.099°
432 88353.709681956 ${\displaystyle D_{3}}$ 0 0 0 24 396 12 0 1290 860 0 9.556°
448 95115.546986209 ${\displaystyle T}$ 0 0 0 24 412 12 0 1338 892 0 9.322°
460 100351.763108673 ${\displaystyle T}$ 0 0 0 24 424 12 0 1374 916 0 9.297°
468 103920.871715127 ${\displaystyle S_{6}}$ 0 0 0 24 432 12 0 1398 932 0 9.120°
470 104822.886324279 ${\displaystyle S_{6}}$ 0 0 0 24 434 12 0 1404 936 0 9.059°

According to a conjecture, if ${\displaystyle m=n+2}$, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p, then the solution for m electrons is f(m): ${\displaystyle f(m)=0^{n}+3n-q}$.[15][clarification needed]

## References

1. ^ Thomson, Joseph John (March 1904). "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" (PDF). Philosophical Magazine. Series 6. 7 (39): 237–265. doi:10.1080/14786440409463107. Archived from the original (PDF) on 13 December 2013.
2. ^ Smale, S. (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX 10.1.1.35.4101. doi:10.1007/bf03025291. S2CID 1331144.
3. ^ Föppl, L. (1912). "Stabile Anordnungen von Elektronen im Atom". J. Reine Angew. Math. 141 (141): 251–301. doi:10.1515/crll.1912.141.251. S2CID 120309200..
4. ^ Schwartz, Richard (2010). "The 5 electron case of Thomson's Problem". arXiv:1001.3702 [math.MG].
5. ^ Yudin, V.A. (1992). "The minimum of potential energy of a system of point charges". Discretnaya Matematika. 4 (2): 115–121 (in Russian).; Yudin, V. A. (1993). "The minimum of potential energy of a system of point charges". Discrete Math. Appl. 3 (1): 75–81. doi:10.1515/dma.1993.3.1.75. S2CID 117117450.
6. ^ Andreev, N.N. (1996). "An extremal property of the icosahedron". East J. Approximation. 2 (4): 459–462. MR1426716, Zbl 0877.51021
7. ^ Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
8. ^ Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
9. ^ a b Batagelj, Vladimir; Plestenjak, Bor. "Optimal arrangements of n points on a sphere and in a circle" (PDF). IMFM/TCS. Archived from the original (PDF) on 25 June 2018.
10. ^ LaFave Jr, Tim (February 2014). "Discrete transformations in the Thomson Problem". Journal of Electrostatics. 72 (1): 39–43. arXiv:1403.2592. doi:10.1016/j.elstat.2013.11.007. S2CID 119309183.
11. ^ Levin, Y.; Arenzon, J. J. (2003). "Why charges go to the Surface: A generalized Thomson Problem". Europhys. Lett. 63 (3): 415. arXiv:cond-mat/0302524. Bibcode:2003EL.....63..415L. doi:10.1209/epl/i2003-00546-1. S2CID 18929981.
12. ^ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
13. ^ LaFave Jr, Tim (2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure". Journal of Electrostatics. 71 (6): 1029–1035. arXiv:1403.2591. doi:10.1016/j.elstat.2013.10.001. S2CID 118480104.
14. ^ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
15. ^ "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.