Image of the LJ135 cluster

Symmetry of the Lennard-Jones clusters

Víctor Luaña

Universidad de Oviedo, Departamento de Química Física y Analítica, c/ Julián Clavería 8, 33006-Oviedo, Spain.

LJ potential

The simple Lennard-Jones pair potential, U(r) = 4 epsilon [(sigma/r)12 - (sigma/r)6], yields an adequate description of the structural and dynamic properties of rare gases. In the case of Argon, for instance, experimental measurements of the second virial coeficient can be fitted to the LJ formula with sigma = 3.405 Å and epsilon/kB = 119.8 K. The potential is easily converted into a universal function by scaling the energies according to the epsilon parameter and the distances by the sigma term. This has helped to the popularity of LJ potential since the very first days of computational modelling of gases, liquids and solids.

LJ clusters

Clusters containing from 3 to about 1000 atoms interacting according to the Universal Lennard-Jones pair potential have been optimized by a number of authors. The Cambridge Cluster Data Base contains the best geometries currently known. We show here the grometry and molecular symmetry of those clusters.

Symmetry

The tessel code has been used to determine and plot the symmetry elements and point group of the cluster. This task uses only the type and cartesian coordinates of each atom in the cluster and works with independence of the global position and orientation of the cluster. The algorithm depends, however, on several internal tolerances, the most important one being the distance tolerance used to decide that two quite close cartesian positions are actually the same point. This tolerance is required because there is always an error in the input geometry of the molecule. The Cambridge Cluster Data Base provides 10-11 significant digits on each cartesian coordinate, but the actual error depends on the optimization level of each cluster. Tolerances of 10-4--10-3 (in units of sigma) are required to converge the symmetry group of some clusters.

Table description

The table below contains the following information relative to each cluster: (1) a small image that can be clicked to obtain a 600x600 raytraced PNG file, (2) the number of atoms, (3) the reduced energy (in epsilon units), (4) a link to a VRML-1 file that can be manipulated on screen, (5) the point group symmetry, (6) a reference to the article containing this geometry, (7-12) symmetry point groups for several distance tolerances.

Other LJ clusters

See also the LJ310-561 and LJ562-1000 cluster symmetry plots.

LJ3-150

Symmetry at several tolerances
LJ Energy VRML Symmetry Reference 3e-5 1e-4 5e-4 1e-3 5e-3 1e-2
Image of the LJ3 cluster 3 -3.000000 file D3h C3h D3h D3h D3h D3h D3h
Image of the LJ4 cluster 4 -6.000000 file Td Td Td Td Td Td Td
Image of the LJ5 cluster 5 -9.103852 file D3h 11 D3h D3h D3h D3h D3h D3h
Image of the LJ6 cluster 6 -12.712062 file Oh 11 Oh Oh Oh Oh Oh Oh
Image of the LJ7 cluster 7 -16.505384 file D5h 11 D5h D5h D5h D5h D5h D5h
Image of the LJ8 cluster 8 -19.821489 file Cs 11 C1 Cs Cs Cs Cs Cs
Image of the LJ9 cluster 9 -24.113360 file C2v 11 C2v C2v C2v C2v C2v C2v
Image of the LJ10 cluster 10 -28.422532 file C3v 11 C3v C3v C3v C3v C3v C3v
Image of the LJ11 cluster 11 -32.765970 file C2v 11 C2v C2v C2v C2v C2v C2v
Image of the LJ12 cluster 12 -37.967600 file C5v 11 - C5v C5v C5v C5v C5v
Image of the LJ13 cluster 13 -44.326801 file Ih 11 Ih Ih Ih Ih Ih Ih
Image of the LJ14 cluster 14 -47.845157 file C3v 11 C3v C3v C3v C3v C3v C3v
Image of the LJ15 cluster 15 -52.322627 file C2v 11 - C2 C2v C2v C2v C2v
Image of the LJ16 cluster 16 -56.815742 file Cs 11 Cs Cs Cs Cs Cs Cs
Image of the LJ17 cluster 17 -61.317995 file C2 9 C1 C2 C2 C2 C2 C2
Image of the LJ18 cluster 18 -66.530949 file C5v 11 Cs - C5v C5v C5v C5v
Image of the LJ19 cluster 19 -72.659782 file D5h 11 D5h D5h D5h D5h D5h D5h
Image of the LJ20 cluster 20 -77.177043 file C2v 11 C1 C1 C2v C2v C2v C2v
Image of the LJ21 cluster 21 -81.684571 file C2v 11 C1 C2v C2v C2v C2v C2v
Image of the LJ22 cluster 22 -86.809782 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ23 cluster 23 -92.844472 file D3h 8 Cs D3h D3h D3h D3h D3h
Image of the LJ24 cluster 24 -97.348815 file Cs 17 C1 C1 Cs Cs Cs Cs
Image of the LJ25 cluster 25 -102.372663 file Cs 11 C1 Cs Cs Cs Cs Cs
Image of the LJ26 cluster 26 -108.315616 file Td 11 C1 - Td Td Td Td
Image of the LJ27 cluster 27 -112.873584 file C2v 14 C1 C1 Cs C2v C2v C2v
Image of the LJ28 cluster 28 -117.822402 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ29 cluster 29 -123.587371 file D3h 11 C2 C2v D3h D3h D3h D3h
Image of the LJ30 cluster 30 -128.286571 file C2v 14 C1 C1 Cs Cs C2v C2v
Image of the LJ31 cluster 31 -133.586422 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ32 cluster 32 -139.635524 file C2v 14 C1 Cs Cs Cs C2v C2v
Image of the LJ33 cluster 33 -144.842719 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ34 cluster 34 -150.044528 file C2v 14 C1 C2v C2v C2v C2v C2v
Image of the LJ35 cluster 35 -155.756643 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ36 cluster 36 -161.825363 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ37 cluster 37 -167.033672 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ38 cluster 38 -173.928427 file Oh 10/15/6 C1 - Oh Oh Oh Oh
Image of the LJ39 cluster 39 -180.033185 file C5v 14 C1 C5v C5v C5v C5v C5v
Image of the LJ40 cluster 40 -185.249839 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ41 cluster 41 -190.536277 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ42 cluster 42 -196.277534 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ43 cluster 43 -202.364664 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ44 cluster 44 -207.688728 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ45 cluster 45 -213.784862 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ46 cluster 46 -220.680330 file C2v 14 C1 C1 C2v C2v C2v C2v
Image of the LJ47 cluster 47 -226.012256 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ48 cluster 48 -232.199529 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ49 cluster 49 -239.091864 file C3v 14 C1 C3 C3v C3v C3v C3v
Image of the LJ50 cluster 50 -244.549926 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ51 cluster 51 -251.253964 file C2v 14 C1 C1 C1 C2v C2v C2v
Image of the LJ52 cluster 52 -258.229991 file C3v 14 Cs C3v C3v C3v C3v C3v
Image of the LJ53 cluster 53 -265.203016 file C2v 14 Cs C2v C2v C2v C2v C2v
Image of the LJ54 cluster 54 -272.208631 file C5v 14 C1 - C5v C5v C5v C5v
Image of the LJ55 cluster 55 -279.248470 file Ih 11 C5v Ih Ih Ih Ih Ih
Image of the LJ56 cluster 56 -283.643105 file C3v 14 C3v C3v C3v C3v C3v C3v
Image of the LJ57 cluster 57 -288.342625 file Cs 14 C1 C1 C1 C1 Cs Cs
Image of the LJ58 cluster 58 -294.378148 file C3v 14 Cs Cs C3v C3v C3v C3v
Image of the LJ59 cluster 59 -299.738070 file C2v 14 C1 C2 C2 C2v C2v C2v
Image of the LJ60 cluster 60 -305.875476 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ61 cluster 61 -312.008896 file C2v 14 Cs C2v C2v C2v C2v C2v
Image of the LJ62 cluster 62 -317.353901 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ63 cluster 63 -323.489734 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ64 cluster 64 -329.620147 file Cs 14 C1 C1 C1 Cs Cs Cs
Image of the LJ65 cluster 65 -334.971532 file C2 18 C1 C1 C2 C2 C2 C2
Image of the LJ66 cluster 66 -341.110599 file C1 4/18 C1 C1 C1 C1 C1 C1
Image of the LJ67 cluster 67 -347.252007 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ68 cluster 68 -353.394542 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ69 cluster 69 -359.882566 file C5v 16/3/12 C5 C5v C5v C5v C5v C5v
Image of the LJ70 cluster 70 -366.892251 file C5v 14 C5v C5v C5v C5v C5v C5v
Image of the LJ71 cluster 71 -373.349661 file C5v 14 C1 C5v C5v C5v C5v C5v
Image of the LJ72 cluster 72 -378.637253 file Cs 4 C1 Cs Cs Cs Cs Cs
Image of the LJ73 cluster 73 -384.789377 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ74 cluster 74 -390.908500 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ75 cluster 75 -397.492331 file D5h 6 C1 D5 D5h D5h D5h D5h
Image of the LJ76 cluster 76 -402.894866 file Cs 6 Cs Cs Cs Cs Cs Cs
Image of the LJ77 cluster 77 -409.083517 file C2v 6 Cs Cs C2v C2v C2v C2v
Image of the LJ78 cluster 78 -414.794401 file Cs 16/3/12 Cs Cs Cs Cs Cs Cs
Image of the LJ79 cluster 79 -421.810897 file C2v 14 C1 C1 C2v C2v C2v C2v
Image of the LJ80 cluster 80 -428.083564 file Cs 14 C1 Cs Cs Cs Cs Cs
Image of the LJ81 cluster 81 -434.343643 file C2v 14 C1 C1 C2v C2v C2v C2v
Image of the LJ82 cluster 82 -440.550425 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ83 cluster 83 -446.924094 file C2v 14 C1 C1 C1 Cs C2v C2v
Image of the LJ84 cluster 84 -452.657214 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ85 cluster 85 -459.055799 file C3v 14 C1 Cs C3v C3v C3v C3v
Image of the LJ86 cluster 86 -465.384493 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ87 cluster 87 -472.098165 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ88 cluster 88 -479.032630 file Cs 5 C1 C1 C1 C1 Cs Cs
Image of the LJ89 cluster 89 -486.053911 file C3v 14 C1 C1 C3v C3v C3v C3v
Image of the LJ90 cluster 90 -492.433908 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ91 cluster 91 -498.811060 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ92 cluster 92 -505.185309 file C3v 14 C1 Cs Cs Cs C3v C3v
Image of the LJ93 cluster 93 -510.877688 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ94 cluster 94 -517.264131 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ95 cluster 95 -523.640211 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ96 cluster 96 -529.879146 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ97 cluster 97 -536.681383 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ98 cluster 98 -543.665361 file Td 13 Td Td Td Td Td Td
Image of the LJ99 cluster 99 -550.666526 file C2v 14 Cs Cs C2v C2v C2v C2v
Image of the LJ100 cluster 100 -557.039820 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ101 cluster 101 -563.411308 file C2v 14 C1 C1 C2v C2v C2v C2v
Image of the LJ102 cluster 102 -569.363652 file C2v 7 C1 C1 C2v C2v C2v C2v
Image of the LJ103 cluster 103 -575.766131 file Cs 7 C1 Cs Cs Cs Cs Cs
Image of the LJ104 cluster 104 -582.086642 file C2v 7 Cs Cs C2v C2v C2v C2v
Image of the LJ105 cluster 105 -588.266501 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ106 cluster 106 -595.061072 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ107 cluster 107 -602.007110 file Cs 16/3/12 C1 C1 Cs Cs Cs Cs
Image of the LJ108 cluster 108 -609.033011 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ109 cluster 109 -615.411166 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ110 cluster 110 -621.788224 file Cs 14 C1 C1 Cs Cs Cs Cs
Image of the LJ111 cluster 111 -628.068416 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ112 cluster 112 -634.874626 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ113 cluster 113 -641.794704 file Cs 3/12 Cs Cs Cs Cs Cs Cs
Image of the LJ114 cluster 114 -648.833100 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ115 cluster 115 -655.756307 file C5v 3/12 C1 C1 C1 C1 C1 C5v
Image of the LJ116 cluster 116 -662.809353 file C5v 14 C5v C5v C5v C5v C5v C5v
Image of the LJ117 cluster 117 -668.282701 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ118 cluster 118 -674.769635 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ119 cluster 119 -681.419158 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ120 cluster 120 -687.021982 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ121 cluster 121 -693.819577 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ122 cluster 122 -700.939379 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ123 cluster 123 -707.802109 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ124 cluster 124 -714.920896 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ125 cluster 125 -721.303235 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ126 cluster 126 -727.349853 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ127 cluster 127 -734.479629 file C2v 14 C2v C2v C2v C2v C2v C2v
Image of the LJ128 cluster 128 -741.332100 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ129 cluster 129 -748.460647 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ130 cluster 130 -755.271073 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ131 cluster 131 -762.441558 file C2v 14 C2v C2v C2v C2v C2v C2v
Image of the LJ132 cluster 132 -768.042203 file C1 14 C1 C1 C1 C1 C1 C1
Image of the LJ133 cluster 133 -775.023203 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ134 cluster 134 -782.206157 file C3v 18 C3v C3v C3v C3v C3v C3v
Image of the LJ135 cluster 135 -790.278120 file Ih 14 Ih Ih Ih Ih Ih Ih
Image of the LJ136 cluster 136 -797.453259 file C5v 14 C5v C5v C5v C5v C5v C5v
Image of the LJ137 cluster 137 -804.631473 file C2v 14 C2v C2v C2v C2v C2v C2v
Image of the LJ138 cluster 138 -811.812780 file C3v 14 C3v C3v C3v C3v C3v C3v
Image of the LJ139 cluster 139 -818.993848 file C2v 14 C2v C2v C2v C2v C2v C2v
Image of the LJ140 cluster 140 -826.174676 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ141 cluster 141 -833.358586 file C5v 14 C5v C5v C5v C5v C5v C5v
Image of the LJ142 cluster 142 -840.538610 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ143 cluster 143 -847.721698 file C2v 14 C2v C2v C2v C2v C2v C2v
Image of the LJ144 cluster 144 -854.904499 file C3v 14 C3v C3v C3v C3v C3v C3v
Image of the LJ145 cluster 145 -862.087012 file C2v 14 C2v C2v C2v C2v C2v C2v
Image of the LJ146 cluster 146 -869.272573 file C5v 14 C5v C5v C5v C5v C5v C5v
Image of the LJ147 cluster 147 -876.461207 file Ih 14 Ih Ih Ih Ih Ih Ih
Image of the LJ148 cluster 148 -881.072971 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ149 cluster 149 -886.693405 file Cs 14 Cs Cs Cs Cs Cs Cs
Image of the LJ150 cluster 150 -893.310258 file C3v 14 C3v C3v C3v C3v C3v C3v

Other LJ clusters

See also the LJ310-561 and LJ562-1000 cluster symmetry plots.

References

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  2. V. Luaña and A. Martín Pendás, Automatic determination of the symmetry elements and point-group symmetry of arbitrary molecules (to be published).
  3. C. Barron, S. Gomez and D. Romero, Appl. Math. Lett., 10 (1997) 25.
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  11. M.R. Hoare and P. Pal, Adv. Phys. 20 (1971) 161; Nature (Physical Sciences) 230 (1971) 5; Nature (Physical Sciences) 236 (1972) 35.
  12. R.H. Leary, J. Global Optimization 11 (1997) 35.
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  14. J.A. Northby, J. Chem. Phys. 87 (1987) 6166.
  15. J. Pillardy and L. Piela, J. Phys. Chem., 99 (1995) 11805.
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  18. G.L. Xue, J. Global Optimization, 4 (1994) 425.





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