Arrangements of Points on a Sphere With Minimal Lennard-Jones Potential

• Library of best ways known to us to distribute n points in 3 dimensions so as to minimize the Lennard-Jones potential
  SUM(i < j) ( dist(P_i, P_j)^(-12) - 2 dist(P_i, P_j)^(-6))

  for n up through 75.

• Copyright 1994 by R. H. Hardin & N. J. A. Sloane

• Reference:
  Minimal-Energy Clusters of Hard Spheres, N. J. A. Sloane, R. H. Hardin, T. S. Duff and J. H. Conway, Discrete Computational Geom., 14 (1995), pp. 237-259. [Note: Fig. 11 , Fig. 12 and "photos" of the putatively optimal clusters of 4 to 10 and 13 to 20 spheres are in separate files]

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• Summary of results:

  Minimum Lennard-Jones Potential

  SUM(i < j) ( dist(P_i, P_j)^(-12) - 2 dist(P_i, P_j)^(-6))

  Copyright R. H. Hardin & N. J. A. Sloane, Feb 1994

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  dim npts LJ potential
  
    3    4  -6.000000000000
    3    5  -9.103852415708
    3    6 -12.712062256809
    3    7 -16.505384168012
    3    8 -19.821489192155
    3    9 -24.113360433647
    3   10 -28.422531893438
    3   11 -32.765970089979
    3   12 -37.967599562364
    3   13 -44.326801419534
    3   14 -47.845156782593
    3   15 -52.322627261839
    3   16 -56.815741780426
    3   17 -61.317994660111
    3   18 -66.530949463140
    3   19 -72.659782454392
    3   20 -77.177042568297
    3   21 -81.684571131370
    3   22 -86.809782259321
    3   23 -92.844472428346
    3   24 -97.348815118023
    3   25 -102.372663199589
    3   26 -108.315616231727
    3   27 -112.873584273756
    3   28 -117.822401723982
    3   29 -123.587371400385
    3   30 -128.286570702954
    3   31 -133.586421919416
    3   32 -139.635523769498
    3   33 -144.842718840540
    3   34 -150.044528410543
    3   35 -155.756643278689
    3   36 -161.825362662323
    3   37 -167.033672486601
    3   38 -173.134317008842
    3   39 -180.033185202882
    3   40 -185.249838615146
    3   41 -190.536277276732
    3   42 -196.277533507971
    3   43 -202.364663584668
    3   44 -207.688727510986
    3   45 -213.784862350209
    3   46 -220.680329909814
    3   47 -226.012256137575
    3   48 -232.199529444526
    3   49 -239.091864233938
    3   50 -244.549926327417
    3   51 -251.253964036603
    3   52 -258.229991073932
    3   53 -265.203016143465
    3   54 -272.208630971240
    3   55 -279.248470463021
    3   56 -283.643105296307
    3   57 -288.342624657286
    3   58 -294.378147633850
    3   59 -299.738069624580
    3   60 -305.875475530358
    3   61 -312.008896093551
    3   62 -317.353900869306
    3   63 -323.489733762676
    3   64 -329.620147123072
    3   65 -334.914685031378
    3   66 -341.110599221148
    3   67 -347.252006894657
    3   68 -353.394542214059
    3   69 -359.726242712763
    3   70 -366.892251224384
    3   71 -373.349660916760
    3   72 -378.637253371079
    3   73 -384.789376967783
    3   74 -390.908499862901
    3   75 -396.282248825571