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 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

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

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 = 2, the optimal configuration consists of electrons at antipodal points.
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 and pentagonal, respectively.

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 $f(x)=x^{-\alpha }$ also known as Riesz $\alpha$ -kernels. For integrable Riesz kernels see ^[7] and for non-integrable Riesz kernels see.^[8] 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.^[9]

"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^[10]

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.^[11]

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, 24) the convex hull is only topologically equivalent to the uniform polyhedron or Johnson solid listed in the last column.^[12]

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	–	–	–	–	–	–	1	–	–	180.000°	digon
3	1.732050808	$D_{{3h}}$	0	–	–	–	–	–	–	3	1	–	120.000°	triangle
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	69.190°	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°	edge-contracted icosahedron
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°

According to a conjecture, if $m=n+2$ , 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): $f(m)=0^{n}+3n-q$ .^[13]

References

↑ 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. Archived from the original (PDF) on 13 December 2013.
↑ 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.
↑ L. Foppl, "Stabile anordnungen von elektronen im atom", J. Reine Angew. Math, 141 (1912), 251–301.
↑ http://arxiv.org/abs/1001.3702
↑ 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
↑ N.N. Andreev, "An extremal property of the icosahedron", East J. Approximation, 2(4) (1996), 459–462, MR 97m:52022, Zbl 0877.51021
↑ 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.
↑ Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
↑ Y. Levin and J. J. Arenzon, ``Why charges go to the Surface: A generalized Thomson Problem Europhys. Lett. Vol. 63 p. 415 (2003)
↑ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
↑ LaFave Jr, Tim (December 2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure" (PDF). Journal of Electrostatics. 71 (6): 1029–1035. doi:10.1016/j.elstat.2013.10.001. Retrieved 10 Feb 2014.
↑ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
↑ "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.

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.
Cris Cecka, Mark J. Bowick, and Alan A. Middleton: http://thomson.phy.syr.edu/
David J. Wales and Sidika Ulker: http://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html and also http://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson2/table.html

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