Best know minima for the Thomson Problem

Previous tables

One of the best prior sources of putative solutions for the Thomson Problem is the Cambridge Cluster Database: Global Minima for the Thomson Problem and Lowest minima located for the Thomson problem. The wikipedia page Thomson Problem (23:56, 30 July 2024) has some that are not in the Cambridge database, and also some improved values, but lacks any reference to its origin. Here are some notes on those tables (mostly the Cambridge database):

n=263: Value given for energy is larger than that computed from their point set. Energy should be 32229.9148956.

n=360: I have found an improved arrangement with D3 symmetry.

n=377: Point set's energy and symmetry (67022.7381628, C1) doesn't match lower energy and symmetry in table.

n=380: Point set's energy of 68109.8721551 is higher than value in table.

n=381: Point set's energy of 68474.1817445 is higher than value in table.

n=382: I have found an improved arrangement with D5 symmetry. (This is the same energy as in the wikipedia page.)

n=388: Symmetry is misidentified as C1. It should be shown as C2v.

n=392: Both Cambridge database and wikipedia page misidentify as C1. It is actually I.

Method

Several procedures were used to construct Table 3-1. In the first method many of the highest IQ results of the constrained search for roundest polyhedra were used as starting points. After optimizing for least energy, the best was selected.

With the second method random arrangements of points on a sphere were used as starting points. This works quite well up to about n=200. For n>300 the number of trials needed gets to be rather large.

For the last method an attempt was made to duplicate the Wales Group's basin-hopping technique. In order for this to work well the optimization routine should attempt to return an arrangement that is in the same basin as the unoptimized points are in.

Steepest descent was used for optimizing, with the step size being chosen by several methods. An estimate of the average expected vornoi cell radius is r = 2/√n. In the first mode the step size is set so that the largest change of position of any point in the set is a fraction of r, typically 0.3*r. In successive iterations the partially optimized set nears the local minimum (descends deeper into the basin), the magnitude of the gradient decreases, and hence the chosen step size increases. When the step size reaches α0 the second mode is initiated: golden section search between α0 and α1. (Currently α0 = 0.35/n1.1, α1 = 9.6*α0, but more testing is needed for values of n not near 400.) As the local minimum is approached (RMS of point changes is less than 0.0001) the step size is fixed at that of the last iteration of the golden section search.

The process starts with n points uniformly distributed on the unit sphere, which is then optimized to the local minimum. At each subsequent step each point is displaced by a random amount limited by a disc with a radius of about r. This set is then optimized and kept if it is better than the previous best.

Table 3-1: Summary min energy for n=4 through 500
nSEpointsvoronoi
4 Td 3.674234614174 e.4.txt e.4.3d
5 D3h 6.474691494688 e.5.txt e.5.3d
6 Oh 9.985281374238 e.6.txt e.6.3d
7 D5h 14.45297741422 e.7.txt e.7.3d
8 D4d 19.67528786123 e.8.txt e.8.3d
9 D3h 25.75998653126 e.9.txt e.9.3d
10 D4d 32.71694946014 e.10.txt e.10.3d
11 C2v 40.59645050819 e.11.txt e.11.3d
12 Ih 49.16525305762 e.12.txt e.12.3d
13 C2v 58.85323061170 e.13.txt e.13.3d
14 D6d 69.30636329662 e.14.txt e.14.3d
15 D3 80.67024411429 e.15.txt e.15.3d
16 T 92.91165530254 e.16.txt e.16.3d
17 D5h 106.0504048286 e.17.txt e.17.3d
18 D4d 120.0844674474 e.18.txt e.18.3d
19 C2v 135.0894675566 e.19.txt e.19.3d
20 D3h 150.8815683337 e.20.txt e.20.3d
21 C2v 167.6416223992 e.21.txt e.21.3d
22 Td 185.2875361492 e.22.txt e.22.3d
23 D3 203.9301906628 e.23.txt e.23.3d
24 O 223.3470740518 e.24.txt e.24.3d
25 Cs 243.8127602987 e.25.txt e.25.3d
26 C2 265.1333263173 e.26.txt e.26.3d
27 D5h 287.3026150330 e.27.txt e.27.3d
28 T 310.4915423582 e.28.txt e.28.3d
29 D3 334.6344399204 e.29.txt e.29.3d
30 D2 359.6039459037 e.30.txt e.30.3d
31 C3v 385.5308380633 e.31.txt e.31.3d
32 Ih 412.2612746505 e.32.txt e.32.3d
33 Cs 440.2040574476 e.33.txt e.33.3d
34 D2 468.9048532813 e.34.txt e.34.3d
35 C2 498.5698724906 e.35.txt e.35.3d
36 D2 529.1224083754 e.36.txt e.36.3d
37 D5h 560.6188877310 e.37.txt e.37.3d
38 D6d 593.0385035664 e.38.txt e.38.3d
39 D3h 626.3890090168 e.39.txt e.39.3d
40 Td 660.6752788346 e.40.txt e.40.3d
41 D3h 695.9167443418 e.41.txt e.41.3d
42 D5h 732.0781075436 e.42.txt e.42.3d
43 C2v 769.1908464591 e.43.txt e.43.3d
44 Oh 807.1742630845 e.44.txt e.44.3d
45 D3 846.1884010610 e.45.txt e.45.3d
46 T 886.1671136391 e.46.txt e.46.3d
47 Cs 927.0592706797 e.47.txt e.47.3d
48 O 968.7134553437 e.48.txt e.48.3d
49 C3 1011.557182653 e.49.txt e.49.3d
50 D6d 1055.182314726 e.50.txt e.50.3d
51 D3 1099.819290318 e.51.txt e.51.3d
52 C3 1145.418964319 e.52.txt e.52.3d
53 C2v 1191.922290416 e.53.txt e.53.3d
54 C2 1239.361474729 e.54.txt e.54.3d
55 C2 1287.772720782 e.55.txt e.55.3d
56 D2 1337.094945275 e.56.txt e.56.3d
57 D3 1387.383229252 e.57.txt e.57.3d
58 D2 1438.618250640 e.58.txt e.58.3d
59 C2 1490.773335278 e.59.txt e.59.3d
60 D3 1543.830400976 e.60.txt e.60.3d
61 C1 1597.941830198 e.61.txt e.61.3d
62 D5 1652.909409898 e.62.txt e.62.3d
63 D3 1708.879681503 e.63.txt e.63.3d
64 D2 1765.802577927 e.64.txt e.64.3d
65 C2 1823.667960263 e.65.txt e.65.3d
66 C2 1882.441525304 e.66.txt e.66.3d
67 D5 1942.122700405 e.67.txt e.67.3d
68 D2 2002.874701748 e.68.txt e.68.3d
69 D3 2064.533483234 e.69.txt e.69.3d
70 D2d 2127.100901550 e.70.txt e.70.3d
71 C2 2190.649906425 e.71.txt e.71.3d
72 I 2255.001190974 e.72.txt e.72.3d
73 C2 2320.633883745 e.73.txt e.73.3d
74 C2 2387.072981838 e.74.txt e.74.3d
75 D3 2454.369689039 e.75.txt e.75.3d
76 C2 2522.674871841 e.76.txt e.76.3d
77 D5 2591.850152353 e.77.txt e.77.3d
78 Th 2662.046474566 e.78.txt e.78.3d
79 Cs 2733.248357478 e.79.txt e.79.3d
80 D4d 2805.355875981 e.80.txt e.80.3d
81 C2 2878.522829664 e.81.txt e.81.3d
82 D2 2952.569675286 e.82.txt e.82.3d
83 C2 3027.528488921 e.83.txt e.83.3d
84 C2 3103.465124430 e.84.txt e.84.3d
85 C2 3180.361442938 e.85.txt e.85.3d
86 C2 3258.211605712 e.86.txt e.86.3d
87 C2 3337.000750014 e.87.txt e.87.3d
88 D2 3416.720196758 e.88.txt e.88.3d
89 C2 3497.439018624 e.89.txt e.89.3d
90 D3 3579.091222722 e.90.txt e.90.3d
91 C2 3661.713699320 e.91.txt e.91.3d
92 D2 3745.291636240 e.92.txt e.92.3d
93 C2 3829.844338421 e.93.txt e.93.3d
94 D2 3915.309269620 e.94.txt e.94.3d
95 C2 4001.771675565 e.95.txt e.95.3d
96 C2 4089.154010055 e.96.txt e.96.3d
97 C2 4177.533599622 e.97.txt e.97.3d
98 C2 4266.822464156 e.98.txt e.98.3d
99 C2 4357.139163131 e.99.txt e.99.3d
100 T 4448.350634331 e.100.txt e.100.3d
101 D3 4540.590051694 e.101.txt e.101.3d
102 D3 4633.736565898 e.102.txt e.102.3d
103 C2 4727.836616833 e.103.txt e.103.3d
104 D6 4822.876522748 e.104.txt e.104.3d
105 D3 4919.000637615 e.105.txt e.105.3d
106 D2 5015.984595704 e.106.txt e.106.3d
107 C2 5113.953547713 e.107.txt e.107.3d
108 C2 5212.813507830 e.108.txt e.108.3d
109 C2 5312.735079920 e.109.txt e.109.3d
110 D6 5413.549294192 e.110.txt e.110.3d
111 D3 5515.293214586 e.111.txt e.111.3d
112 D5 5618.044882326 e.112.txt e.112.3d
113 D3 5721.824978027 e.113.txt e.113.3d
114 C2 5826.521572162 e.114.txt e.114.3d
115 C3 5932.181285777 e.115.txt e.115.3d
116 C2 6038.815593578 e.116.txt e.116.3d
117 C2 6146.342446578 e.117.txt e.117.3d
118 C2 6254.877027789 e.118.txt e.118.3d
119 C2 6364.347317479 e.119.txt e.119.3d
120 Cs 6474.756324979 e.120.txt e.120.3d
121 C3 6586.121949584 e.121.txt e.121.3d
122 Ih 6698.374499260 e.122.txt e.122.3d
123 C2v 6811.827228174 e.123.txt e.123.3d
124 D2 6926.169974193 e.124.txt e.124.3d
125 C2 7041.473264023 e.125.txt e.125.3d
126 D4 7157.669224866 e.126.txt e.126.3d
127 D5 7274.819504675 e.127.txt e.127.3d
128 C2 7393.007443068 e.128.txt e.128.3d
129 C2 7512.107319268 e.129.txt e.129.3d
130 C2 7632.167378912 e.130.txt e.130.3d
131 C2 7753.205166940 e.131.txt e.131.3d
132 I 7875.045342797 e.132.txt e.132.3d
133 C3 7998.179212898 e.133.txt e.133.3d
134 C2 8122.089721194 e.134.txt e.134.3d
135 D3 8246.909486992 e.135.txt e.135.3d
136 T 8372.743302538 e.136.txt e.136.3d
137 D5 8499.534494781 e.137.txt e.137.3d
138 C2 8627.406389880 e.138.txt e.138.3d
139 C2 8756.227056952 e.139.txt e.139.3d
140 C1 8885.980609040 e.140.txt e.140.3d
141 C2v 9016.615349189 e.141.txt e.141.3d
142 C2 9148.271579993 e.142.txt e.142.3d
143 C2 9280.839851192 e.143.txt e.143.3d
144 D2 9414.371794459 e.144.txt e.144.3d
145 Cs 9548.928837231 e.145.txt e.145.3d
146 D2 9684.381825574 e.146.txt e.146.3d
147 C2 9820.932378373 e.147.txt e.147.3d
148 C2 9958.406004269 e.148.txt e.148.3d
149 C1 10096.85990739 e.149.txt e.149.3d
150 T 10236.19643670 e.150.txt e.150.3d
151 C2 10376.57146927 e.151.txt e.151.3d
152 D2 10517.86759287 e.152.txt e.152.3d
153 D3 10660.08274823 e.153.txt e.153.3d
154 C2 10803.37242114 e.154.txt e.154.3d
155 C2 10947.57469227 e.155.txt e.155.3d
156 C2 11092.79831145 e.156.txt e.156.3d
157 C2 11238.90304115 e.157.txt e.157.3d
158 C2 11385.99018619 e.158.txt e.158.3d
159 C2 11534.02396095 e.159.txt e.159.3d
160 D2 11683.05480554 e.160.txt e.160.3d
161 C2 11833.08473946 e.161.txt e.161.3d
162 D3 11984.05033581 e.162.txt e.162.3d
163 C2 12136.01305322 e.163.txt e.163.3d
164 D2 12288.93010531 e.164.txt e.164.3d
165 C2 12442.80445137 e.165.txt e.165.3d
166 D2d 12597.64907132 e.166.txt e.166.3d
167 C2 12753.46942974 e.167.txt e.167.3d
168 D3 12910.21267226 e.168.txt e.168.3d
169 Cs 13068.00645112 e.169.txt e.169.3d
170 D2d 13226.68107854 e.170.txt e.170.3d
171 D3 13386.35593071 e.171.txt e.171.3d
172 C2v 13547.01810878 e.172.txt e.172.3d
173 Cs 13708.63524303 e.173.txt e.173.3d
174 D2 13871.18709229 e.174.txt e.174.3d
175 C2 14034.78130692 e.175.txt e.175.3d
176 C1 14199.35477563 e.176.txt e.176.3d
177 D5 14364.83754529 e.177.txt e.177.3d
178 D2 14531.30955258 e.178.txt e.178.3d
179 C1 14698.75459422 e.179.txt e.179.3d
180 D2 14867.09992752 e.180.txt e.180.3d
181 C2 15036.46723976 e.181.txt e.181.3d
182 D5 15206.73061090 e.182.txt e.182.3d
183 C1 15378.16657102 e.183.txt e.183.3d
184 T 15550.42145031 e.184.txt e.184.3d
185 C2 15723.72007407 e.185.txt e.185.3d
186 C1 15897.89743704 e.186.txt e.186.3d
187 D5 16072.97518632 e.187.txt e.187.3d
188 D2 16249.22267887 e.188.txt e.188.3d
189 C2 16426.37193886 e.189.txt e.189.3d
190 C3 16604.42833850 e.190.txt e.190.3d
191 C1 16783.45221936 e.191.txt e.191.3d
192 I 16963.33838646 e.192.txt e.192.3d
193 C2 17144.56474087 e.193.txt e.193.3d
194 C1 17326.61613647 e.194.txt e.194.3d
195 D3 17509.48930393 e.195.txt e.195.3d
196 C2 17693.46054808 e.196.txt e.196.3d
197 D5 17878.34016257 e.197.txt e.197.3d
198 C2 18064.26217719 e.198.txt e.198.3d
199 C1 18251.08249563 e.199.txt e.199.3d
200 D2 18438.84271753 e.200.txt e.200.3d
201 C1 18627.59122624 e.201.txt e.201.3d
202 D5 18817.20471826 e.202.txt e.202.3d
203 Cs 19007.98120458 e.203.txt e.203.3d
204 Th 19199.54077560 e.204.txt e.204.3d
205 C1 19392.36915238 e.205.txt e.205.3d
206 C2 19585.95585654 e.206.txt e.206.3d
207 C2 19780.65690931 e.207.txt e.207.3d
208 C2 19976.20326120 e.208.txt e.208.3d
209 C2 20172.75468066 e.209.txt e.209.3d
210 C2 20370.25161512 e.210.txt e.210.3d
211 C1 20568.74060277 e.211.txt e.211.3d
212 I 20768.05308596 e.212.txt e.212.3d
213 C1 20968.61202549 e.213.txt e.213.3d
214 T 21169.91041037 e.214.txt e.214.3d
215 C1 21372.34878934 e.215.txt e.215.3d
216 D3 21575.59637786 e.216.txt e.216.3d
217 D5 21779.85608041 e.217.txt e.217.3d
218 C2 21985.26394892 e.218.txt e.218.3d
219 C2 22191.48547481 e.219.txt e.219.3d
220 D2 22398.65560253 e.220.txt e.220.3d
221 C2 22606.88154725 e.221.txt e.221.3d
222 D3 22816.02557024 e.222.txt e.222.3d
223 C2 23026.16588116 e.223.txt e.223.3d
224 D2 23237.24487348 e.224.txt e.224.3d
225 C1 23449.43646066 e.225.txt e.225.3d
226 C3 23662.51112265 e.226.txt e.226.3d
227 C2 23876.57689371 e.227.txt e.227.3d
228 C2 24091.57898586 e.228.txt e.228.3d
229 C1 24307.59931328 e.229.txt e.229.3d
230 C2 24524.48535094 e.230.txt e.230.3d
231 C2 24742.38249476 e.231.txt e.231.3d
232 T 24961.25231893 e.232.txt e.232.3d
233 D3 25181.05739953 e.233.txt e.233.3d
234 C2 25401.93178664 e.234.txt e.234.3d
235 C1 25623.76314420 e.235.txt e.235.3d
236 C2 25846.50056315 e.236.txt e.236.3d
237 C1 26070.36701545 e.237.txt e.237.3d
238 C2 26295.12604790 e.238.txt e.238.3d
239 C1 26520.87411744 e.239.txt e.239.3d
240 C2 26747.50820509 e.240.txt e.240.3d
241 C3 26975.19028397 e.241.txt e.241.3d
242 D3 27203.79928555 e.242.txt e.242.3d
243 C2 27433.36735237 e.243.txt e.243.3d
244 D2 27663.90511218 e.244.txt e.244.3d
245 C1 27895.54074539 e.245.txt e.245.3d
246 D2 28128.05146431 e.246.txt e.246.3d
247 C2 28361.53277726 e.247.txt e.247.3d
248 C2 28596.05210294 e.248.txt e.248.3d
249 C2 28831.47390985 e.249.txt e.249.3d
250 C2 29067.87767391 e.250.txt e.250.3d
251 C1 29305.23386162 e.251.txt e.251.3d
252 C2 29543.52286812 e.252.txt e.252.3d
253 C1 29782.91736439 e.253.txt e.253.3d
254 C1 30023.22286183 e.254.txt e.254.3d
255 D3 30264.42425128 e.255.txt e.255.3d
256 T 30506.68751584 e.256.txt e.256.3d
257 D5 30749.94141734 e.257.txt e.257.3d
258 C2 30994.21357748 e.258.txt e.258.3d
259 C3 31239.44230686 e.259.txt e.259.3d
260 D3 31485.57819177 e.260.txt e.260.3d
261 C2 31732.74278773 e.261.txt e.261.3d
262 C1 31980.85182386 e.262.txt e.262.3d
263 D3 32229.91489564 e.263.txt e.263.3d
264 D2 32479.90814117 e.264.txt e.264.3d
265 C2 32731.01239746 e.265.txt e.265.3d
266 C2 32982.98362908 e.266.txt e.266.3d
267 C2 33235.96206603 e.267.txt e.267.3d
268 D2 33489.87448494 e.268.txt e.268.3d
269 C1 33744.80073277 e.269.txt e.269.3d
270 C2 34000.64849701 e.270.txt e.270.3d
271 Cs 34257.44173814 e.271.txt e.271.3d
272 Ih 34515.19329268 e.272.txt e.272.3d
273 D3 34774.25785094 e.273.txt e.273.3d
274 D2 35034.03346670 e.274.txt e.274.3d
275 C2 35294.89405121 e.275.txt e.275.3d
276 D3 35556.63330357 e.276.txt e.276.3d
277 C2 35819.34081095 e.277.txt e.277.3d
278 C2 36083.03689375 e.278.txt e.278.3d
279 C2 36347.65842572 e.279.txt e.279.3d
280 C2 36613.26959007 e.280.txt e.280.3d
281 C1 36879.80561534 e.281.txt e.281.3d
282 I 37147.29441846 e.282.txt e.282.3d
283 C3 37416.07574278 e.283.txt e.283.3d
284 C2 37685.68979324 e.284.txt e.284.3d
285 C2 37956.23945598 e.285.txt e.285.3d
286 C2 38227.67332054 e.286.txt e.286.3d
287 C2 38500.13433228 e.287.txt e.287.3d
288 D3 38773.58128368 e.288.txt e.288.3d
289 C2 39048.04918673 e.289.txt e.289.3d
290 C1 39323.41006203 e.290.txt e.290.3d
291 D3 39599.71181683 e.291.txt e.291.3d
292 D5 39877.00801290 e.292.txt e.292.3d
293 D3 40155.34907602 e.293.txt e.293.3d
294 C2 40434.76229748 e.294.txt e.294.3d
295 C2 40715.14171670 e.295.txt e.295.3d
296 C2 40996.41390852 e.296.txt e.296.3d
297 C1 41278.70329301 e.297.txt e.297.3d
298 C1 41561.92076879 e.298.txt e.298.3d
299 C2 41846.08118343 e.299.txt e.299.3d
300 C1 42131.26413724 e.300.txt e.300.3d
301 C2v 42417.35662983 e.301.txt e.301.3d
302 C1 42704.49375848 e.302.txt e.302.3d
303 C1 42992.52896687 e.303.txt e.303.3d
304 D2 43281.51336496 e.304.txt e.304.3d
305 C2 43571.55045368 e.305.txt e.305.3d
306 Th 43862.56978079 e.306.txt e.306.3d
307 Cs 44154.54770595 e.307.txt e.307.3d
308 D2 44447.45305635 e.308.txt e.308.3d
309 C1 44741.56428380 e.309.txt e.309.3d
310 C2 45036.49857222 e.310.txt e.310.3d
311 C2 45332.40210550 e.311.txt e.311.3d
312 C2 45629.31380400 e.312.txt e.312.3d
313 C2 45927.19254703 e.313.txt e.313.3d
314 D2 46225.97975757 e.314.txt e.314.3d
315 D3 46525.82564343 e.315.txt e.315.3d
316 C2 46826.58840288 e.316.txt e.316.3d
317 D5 47128.31034452 e.317.txt e.317.3d
318 D3 47431.05602004 e.318.txt e.318.3d
319 C1 47734.81376397 e.319.txt e.319.3d
320 C2 48039.46893016 e.320.txt e.320.3d
321 C1 48345.16095278 e.321.txt e.321.3d
322 D2 48651.76218239 e.322.txt e.322.3d
323 C1 48959.40031431 e.323.txt e.323.3d
324 D2 49267.92503237 e.324.txt e.324.3d
325 C2 49577.57669331 e.325.txt e.325.3d
326 C1 49888.08821814 e.326.txt e.326.3d
327 C2 50199.57141959 e.327.txt e.327.3d
328 D2 50512.07305659 e.328.txt e.328.3d
329 C2 50825.60602455 e.329.txt e.329.3d
330 D3 51140.05136022 e.330.txt e.330.3d
331 C2 51455.47545086 e.331.txt e.331.3d
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Wayne Deeter - wrd@deetour.net

Last modified: November 9, 2024